text:
WHEN LAWYERS' CONFLICTS BECOME ALLEGED MALPRACTICE CLAIMS.
Defense counsel retained by liability insurance companies to represent policyholders face potential conflicts of interest sometimes. Whether and when these potential conflicts ripen into claims for damages depends on the law of the jurisdiction and the facts of the case.
In Arden v. Forsberg & Umlauf, P.S., ___ Wash. 2d ___, 402 P.3d 245 (2017), the Supreme Court of Washington held that the policyholders-clients could not claim damages in that case from a retained law firm's alleged breach of duty when the law firm failed to disclose an alleged conflict based on a previous relationship between the firm and the policyholders' liability carrier.
HEALTH CARE COVERAGE.
If you or your clients do not have health insurance coverage, check out and sign up at www.healthcare.gov.
You may not have heard the word that health care coverage is available -- right now, today, even as you read these words. It's your right. It's the law of the land!
Please Read The Disclaimer. ©2017 by Dennis J. Wall. All Rights Reserved.
Posted by Dennis J. Wall, Attorney At Law, a P.A. on October 11, 2017 at 09:12 AM in Fiduciary Duties, Professional Liabilty | Permalink | Comments (0)
Tags: claim, client, conflict, coverage, defense, fiduciary, healthcare, insurance, liability, malpractice, policyholder, professional, retained, Washington
Advance Retainer Held Not Covered Under Law Firm's E&O: Pt. 2.
The first part of this article was posted here on Monday, July 13, 2015. This installment continues and completes this article.
After comparing the allegations of the underlying complaint against the insured, with the provisions of the insurance policy at issue as required by Illinois law as in most jurisdictions, the trial court decided that there was no duty to defend in this matter. As noted, the Illinois First District Court of Appeal affirmed that decision. Illinois State Bar Ass'n Mut. Ins. Co. v. Coleman Law Firm, ¶ 24, 2014 Ill. App. (1st) 133518-U, ¶ 24, 2014 WL 7446203, at *5 (Ill. 1st DCA December 29, 2014)(stated "UNPUBLISHED OPINION"). "Comparing the allegations of the FDIC complaint in the underlying action to the provisions of the insurance policy at hand, the factual allegations in the underlying complaint did not allege acts of negligence, but rather acts which could only be classified as intentional and, thus, excluded from policy coverage." Illinois State Bar Ass'n Mut. Ins. Co. v. Coleman Law Firm, ¶ 36, 2014 Ill. App. (1st) 133518-U, ¶ 36, 2014 WL 7446203, at *9 (Ill. 1st DCA December 29, 2014). [Emphasis added.]
As can readily be seen from the underscored words in this quoted holding, the Illinois appellate court compared the factual allegations, not simply the claims alleged in the underlying complaint, to the provisions of the insurance policy in order to determine whether the policy provided a duty to defend in this case.
Reprinted with the permission of Thomson Reuters and the author from the manuscript of the author's 2015 Supplement chapters, and particularly Section 3:51, in "Litigation and Prevention of Insurer Bad Faith, 3d" ©2015 by Thomson Reuters.
Posted by Dennis J. Wall, Attorney At Law, a P.A. on July 15, 2015 at 06:28 AM in Professional Liabilty | Permalink
Tags: coverage, E&O, law firm, malpractice
In an officially unpublished opinion, Illinois' First District Court of Appeal held that an advance retainer which a law firm required for future costs and expenses was not covered malpractice under the firm's E&O insurance policy: Illinois State Bar Ass'n Mut. Ins. Co. v. Coleman Law Firm, 2014 Ill. App. (1st) 133518-U, 2014 WL 7446203 (Ill. 1st DCA December 29, 2014)(stated "UNPUBLISHED OPINION").
In that case, the Court reported, a bank's "inside directors" signed an "'advanced payment retainer agreement'" with the insured law firm. The retainer agreement was stated to be "in connection with legal matters arising out of [the directors'] capacity as inside directors of the Bank." The amount of the advance retainer was $150,000.00. That amount was, according to the agreement, to be paid by the Bank to the law firm "for reasonable costs and expenses incurred in connection with the matters for which the [Bank's inside directors] had retained Coleman Law". Illinois State Bar Ass'n Mut. Ins. Co. v. Coleman Law Firm, ¶ 4, 2014 Ill. App. (1st) 133518-U, ¶ 4, 2014 WL 7446203, at *1 (Ill. 1st DCA December 29, 2014)(stated "UNPUBLISHED OPINION").
The bank went into receivership. The FDIC was appointed receiver and sent a letter to the law firm demanding the return of the $150,000.00 paid by the bank. The law firm refused. The FDIC filed suit for the return of the money and the law firm tendered defense of the FDIC case to the firm's professional liability insurance carrier.
The carrier in turn filed an action for declaratory relief as to whether there was coverage under the malpractice insurance policy. The Circuit Court in and for Cook County granted the carrier's motion for summary judgment. The First District affirmed on appeal, in the unpublished decision under discussion.
To be continued on Insurance Claims and Issues Weblog.
Reprinted with the permission of Thomson Reuters, and the author, from the manuscript of the author's 2015 Supplement chapters, and in particular Section 3:51, in "Litigation and Prevention of Insurer Bad Faith, 3d" ©2015 by Thomson Reuters.
Tags: coverage, E&O, malpractice
PRIVATIZING PRISONS FOR PROFIT? REPORTS ....
Who has the Coverage?
Reports of despicable conditions in for-profit prisons contain opinions. These reports also contain evidence of facts. Photographs and eyewitnesses have recorded these conditions in a private prison in Mississippi recently:
"Dirt, feces and, occasionally, blood are caked on the walls of cells."
Prisoners with treatable medical conditions, like glaucoma, do not receive medical treatment. When these individuals are released, their condition has deteriorated to the point where the condition is untreatable.
Guards, who are paid a small hourly wage, spend little time in the solitary-confinement block. Prisoners in solitary confinement sometimes set fires to their bedding in order to get the guards' attention and, even then, the guards sometimes do not respond.
These eyewitness accounts and photographic evidence and more are reported for example by Erica Goode, "Seeing Squalor and Unconcern in Southern Jail," p. 1, col. 1 (New York Times Nat'l ed., Sunday, June 8, 2014). The Mississippi prison in question is run by Management and Training Corporation from Utah.
Management and Training Corporation made one comment for publication. They blamed these problems on the corporation they replaced in 2012, one GEO Group. Otherwise, they declined to comment, saying they would not comment because they have been sued. This is unusual. Even aside from the fact that due diligence probably revealed these conditions before Management and Training Corporation took over the running of this Mississippi prison, this is 2014. It has now been two years since CEO Group was in charge, and this is still unusual. Many large corporations stage large press campaigns to "push back" against lawsuits.
Management and Training Corporation is entitled to a defense. Questions abound about how they will defend themselves. Which of their carriers will fund their defense? What sort of insurance coverage will provide them indemnity if their defense fails? Who will cover these people against such reports? And what sort of exclusions could apply, intentional damages exclusions for example? So much depends on what Management and Training Corporation has to say about its prison-keeping. When they start talking we will know more. Until then, questions continue to pile up.
© 2014 by Dennis J. Wall. All rights reserved. No claim to original U.S. Government works.
Allocation, Indemnity, Defense Insurance Coverage Issues: Part Two.
Seeing Squalor and Unconcern in Southern Jail
Posted by Dennis J. Wall, Attorney At Law, a P.A. on June 09, 2014 at 08:57 AM in "Occurrence", Current Affairs, Discovery, Duty to Defend, Evidence, Exclusions, Interpretation and Application of Insurance Contracts, Professional Liabilty, Risk Management | Permalink
SAY IT AIN'T SO, SHEILA: THE FDIC'S SECRET SETTLEMENTS.
I am currently reading with interest Sheila C. Bair's book, "Bull By The Horns". It is a description of the time spent in office by the Federal Deposit and Insurance Corporation's former Chairman, written by the Chairman herself to describe her office. During her tenure, which ended on or about July 6, 2009, much of the Financial Crisis exploded on our economy.
While I am reading Ms. Bair's account of her tenure, reports have surfaced concerning secret FDIC settlements made on her watch with Directors and Officers who allegedly caused or contributed to the cause of the Great Financial Fiasco. In apparent violation of Federal law, the FDIC agreed to keep the settlements secret unless a specific request was made for disclosure of that particular settlement. See, e.g., E. Scott Reckard, "FDIC Begins to Reveal Settlements Related to Financial Crisis" (Los Angeles Times Online, posted March 19, 2013); William K. Black, "Business Blog / Which Aspect of the FDIC's Litigation Failures is The Most Embarrassing and Damaging?" (Huffington Post, posted March 12, 2013). Reportedly, Ms. Bair was leading the FDIC "when most of the settlements were completed". See Scott Reckard, Los Angeles Times, supra.
Sheila Bair does not mention the fact of FDIC settlements with Directors and Officers of failed banks in her book, let alone that the FDIC agreed to keep them secret, or at least I cannot find any mention of them in her book. We shall now, perhaps, all hear her side of the story soon.
In response to a recently published newspaper report in the Los Angeles Times, the FDIC has begun posting information concerning its previously secret settlements on its web site. The documents will be posted in increments. As of March 29, 2013, the initial postings consist of about 45 sets of documents (not simply individual settlements, but in some cases, groups of settlements). They are collectively labeled by the FDIC as "Professional Liability Agreements by State Since 2008". The FDIC site also contains a list of 53 "Professional Liability Lawsuits" filed by the FDIC against "officers and directors, attorneys, accountants, appraisers, brokers or others" and "fidelity bond carriers and title insurance companies," between July 2, 2010 and March 7, 2013.
Examining the 45 pdf's posted to date will take awhile. If I find anything 'newsworthy' in them, I will post my reviews of these settlement documents as I am able to complete my reviews. In the meantime, the settlement which has attracted the most popular interest at the present time is the December, 2012 settlement between the FDIC and Michael Perry, who was CEO of failed IndyMac Bank located in California. IndyMac's failure "cost the [FDIC] $13 billion -- the biggest loss in its 80-year history." Scott Reckard, Los Angeles Times, supra. [Emphasis added.]
The FDIC reportedly secretly settled its claims against Mr. Perry for his agreement to personally pay $1 Million, plus an agreement by Mr. Perry not to engage in the banking industry (for awhile, at least) and also with provisions guaranteeing the FDIC's pursuit of IndyMac's "liability insurers" for $11 Million more. Professor William K. Black, Huffington Post, supra.
The Perry settlement was so good that it was announced in EMails sent to news organizations -- reportedly by Mr. Perry's lawyers, not by the FDIC. The terms of this particular settlement were therefore public knowledge in December, 2012 when the settlement was reached between Mr. Perry and the FDIC. See Kevin LaCroix, "IndyMac CEO Settles FDIC's Failed Bank Suit" (The D & O Diary, posted December 17, 2012).
In contrast, one alleged seller and/or deliverer of Mortgages to IndyMac Bank, Quicken, agreed to pay 6.5 times as much money, or $6.5 Million, to settle the FDIC's claims against it, and "[w]ithout admitting liability". (See the 65th page in the 230 page pdf labeled, "IndyMac2", among the FDIC's "Professional Liability Settlement Agreements by State Since 2008".)
On the whole, it seems less expensive to be a failed CEO than it does to be a failed Mortgage 'seller and/or deliverer'. On the whole, as well, there are many legal entanglements with secret settlements that make these agreements unwise, and the price of keeping them secret exorbitant. This is a good rule of thumb to follow when making all types of insurance settlements, it seems: As a general rule at least, keep them open and transparent. See generally DENNIS J. WALL, 1 LITIGATION AND PREVENTION OF INSURER BAD FAITH § 3:107, "Settlement of Third Party Bad Faith Claims: Confidentiality (protected) or Concealment (Void)" (West Third Edition; 2013 Supplement in process); 2 id. § 9:28, "Settlement of First Party Bad Faith Claims: Confidentiality (Protected) or Concealment (Void)".
Posted by Dennis J. Wall, Attorney At Law, a P.A. on April 01, 2013 at 04:47 AM in Current Affairs, Directors and Officers (D&O) Coverage, Market Performance, Professional Liabilty, Settlement, Title Insurance | Permalink
Tags: http://www.dandodiary.com/admin/trackback/291387
FLORIDA DUTY TO DEFEND INTENTIONAL TORT UNLESS "INTENTIONAL DAMAGES" EXCLUSION ....
.... AND CERTAINLY NOT WHEN THERE ISN'T EVEN ONE IN THE POLICY.
In Philadelphia Indemnity Insurance Co. v. Hamic, 2012 WL 5055558 (M.D. Fla. October 18, 2012), a Federal District Judge held that under Florida Insurance Law, there is a Duty to Defend a claimed Conspiracy to commit Malicious Prosecution.
The Federal Court previously determined that there is a Duty to Defend a Malicious Prosecution Claim in the same underlying case. An Attorney's Professional Liability Policy covering specified Errors and Omissions is at issue in the Federal case. The Defendants in the underlying case, and also the Insureds on the Policy, are an attorney and his Law Firm.
Philadelphia, the E & O Carrier, filed a Motion to Reconsider or to Amend the Summary Judgment of a Duty to Defend, specifically as to alleged civil conspiracy. The Insurance Company argued that conspiracy is an intentional tort and is not covered by the Attorney's Professional Liability Policy at issue.
The Professional Liability Policy at issue has no Exclusion for intentional torts, the Court observed. Philadelphia Indemnity Insurance Co. v. Hamic, 2012 WL 5055558 *1 (M.D. Fla. October 18, 2012).
Moreover, the Professional Liability Policy at issue covers Errors & Omissions some of which are intentional torts, "though not specifically malicious prosecution or conspiracies." Philadelphia Indemnity Insurance Co. v. Hamic, 2012 WL 5055558 *1 (M.D. Fla. October 18, 2012).
Even though conspiracy was claimed in the underlying case, conspiracy does not require an intent to cause the damages complained of, in order to allege it or to prove it. Without an Exclusion of intent to cause the damages complained of, there is no doubt that there is a Duty to Defend such claims under settled Florida Insurance Law. "The tort of civil conspiracy does not require a finding of specific intent to harm the plaintiff or to extract an intended result. Essentially the intent to commit the tort does not also mandate an intent to do the harm that resulted, which was an arrest." Philadelphia Indemnity Insurance Co. v. Hamic, 2012 WL 5055558 *2 (M.D. Fla. October 18, 2012), CITING LIME TREE VILLAGE COMMUNITY CLUB ASS'N v. STATE FARM GENERAL INSURANCE CO., 980 F.2d 1402, 1407 (11th Cir. 1993), in which the Author successfully represented a Florida Homeowner's Association when it and its elderly members were all denied a Defense under their Business Liability Coverage and under their Directors and Officers Liability Coverage, which was upheld on a Summary Judgment by a Federal District Judge, and which was unanimously reversed on appeal by the Eleventh Circuit in the decision relied on by other Federal District Judges to determine a Duty to Defend, including in this October 18, 2012 decision.
Posted by Dennis J. Wall, Attorney At Law, a P.A. on October 22, 2012 at 09:44 AM in Directors and Officers (D&O) Coverage, Duty to Defend, Homeowners' Associations, Professional Liabilty | Permalink | Comments (0) | TrackBack (0)
Competing Deposition Notices of Rule 30(b)(6) Corporate Representatives.
In Diamond State Insurance Co. v. His House, Inc., Download Diamond State Insurance Co. v. His House Inc. (S.D. Fla. Case No. 10.20039.CIV, Discovery Order, Filed Jan. 18, 2011) PUBLIC ACCESS also published as 2011 WL 146837 (S.D. Fla. January 18, 2011), a United States Magistrate Judge was confronted with competing Rule 30(b)(6) Notice of Deposition of Corporate Representatives. One Notice was served on behalf of a Policyholder-Defendant. The other Notice was served on behalf of a Liability Insurance Company-Plaintiff which had already denied all coverage for defense and for indemnity of an underlying lawsuit filed against the Policyholder.
The Nature of the Case: A Declaratory Judgment Action With No Claim For Bad Faith Alleged.
The case began when Diamond State filed suit for declaratory relief against His House and others. His House was a nonprofit corporation engaged in the business of caring for foster care children placed with His House by the State of Florida.
Having already denied all coverage to His House for defense or indemnity with respect to an underlying wrongful death suit, in this case Diamond State sought declarations that (1) a Professional Liability Policy and (2) a Commercial General Liability Policy which it issued to His House did not provide coverage for the wrongful death lawsuit against His House. Id. at *1.
In addition to requesting judicial declarations that the two Policies did not provide coverage under their clear language, Diamond State also alleged that His House's coverage if any under the Professional Liability Policy at issue was voided because His House allegedly "knew of the death of the infant prior to ... the application for the professional liability policy". As to the CGL Policy, Diamond State also alleged that the "occurrence" that caused the child's death "occurred prior to the policy period and thus no coverage is available under that policy either". Id. at *1.
The wrongful death lawsuit or underlying case was filed against His House as the result of the death of an infant, "M.C.," after the child was placed in the care of His House's employee, Ms. Kristen Glaspy, a "foster care mother who was employed by His House". Id. at *1.
The Competing Rule 30(b)(6) Deposition Notices and Motions to Compel: First, His House's Motion to Compel the Deposition of Diamond State's Corporate Representative.
The Policyholder, His House, wanted to depose Diamond State Insurance Company's Corporate Representative on five (5) designated areas, four of them concerning interpretation of the "policies at issue in this case" in one respect or another, and the fifth area concerning the Insurance Company's handling and denial of His House's claim to coverage under the Diamond State Policies. Id. at *3.
Diamond State wanted to re-depose His House's Corporate Representative after His House's Rule 30(b)(6) Deposition had already been taken once in the Declaratory Judgment Action, and once in the underlying case. In addition, Diamond State wanted to depose Ms. Glaspy, who again was an employee of His House, and who had already been deposed in the underlying case as well. The Magistrate Judge resolved the issues surrounding a Rule 30(b)(6) Deposition of the Insurance Company first.
Florida substantive law "governs the substantive portions of this declaratory action," she pointed out. Under Florida law, "the construction of an insurance policy is a question of law to be determined by the court." If the insurance policies were held to be unambiguous, they should be interpreted in accordance with their clear terms. If the Court found the insurance contracts ambiguous, then the Court would be required to construe the wording most strictly against the Insurance Company. Id. at *2.
Florida substantive law therefore made interrogation about the Policies at issue "irrelevant to the resolution of this declaratory action." Id. at *4. If the Policy provisions at issue were unambiguous, there would be no need for testimony about them. If the Court held the policy provisions ambiguous instead, then in this case, at least, "Defendants will automatically prevail in the instant declaratory action." Id. "Thus, under either scenario, the testimony of Diamond's corporate representative is irrelevant." Id.
As to the claim handling inquiry, it might be appropriate in a bad faith case, the Court suggested, see id. at *4-*5, but it was not relevant to this action for declaratory relief concerning the wording used in two insurance policies. As to the allegations made by Diamond State to the effect that His House knew of the infant's death before applying for the Professional Liability Policy, the prospective testimony was again irrelevant. In this case, His House did not dispute the date of the infant's death or the date of its Professional Liability Insurance application. Under the circumstances of this peculiar case, the Court denied His House's Motion to Compel the deposition of Diamond's corporate representative. Id. at *6.
The Online Docket of this case on Pacer, the Online Docket for Federal Courts, reflects that the District Judge granted the Plaintiff Liability Insurance Company's Motion for Summary Judgment on January 31, 2011 or within two weeks after the U.S. Magistrate Judge entered the Discovery Order which is the subject of this post. Download Diamond State Insurance Co. v. His House Inc. (S.D. Fla. Case No. 10.20039.CIV, Order Granting Summary Judgment, Filed Jan. 31, 2011) PUBLIC ACCESS. To be continued ......
Posted by Dennis J. Wall, Attorney At Law, a P.A. on April 25, 2011 at 07:09 AM in Commercial General Liability ("CGL") Insurance, Declaratory Judgment Actions, Discovery, Duty to Defend, Professional Liabilty | Permalink | Comments (0) | TrackBack (1)
In Camera Inspection by U.S. Magistrate Judge: Held, All Documents Privileged.
The case of Carter v. Ozoeneh, Download Carter v. Ozoeneh (W.D.N.C. Case No. 3.2008cv614, Memorandum and Opinion Filed July 22, 2010) FREE ACCESS TO THIS OFFICIAL REPORT also published as 2010 WL 2889582 (W.D.N.C. July 22, 2010)(Westlaw subscription required to access Westlaw) is unique. At least, it is not in the garden variety of most Federal Cases that is certain.
Plaintiff Ronald Carter sued over a patent. He claimed to be the sole inventor of certain technology involved in the patent. He and his corporation sued his lawyers, the Clements Walker Law Firm, and various persons (apparently almost all but not quite all of whom are lawyers who represented him regarding the patent), for alleged malpractice in connection with the patent. It is confusing, but apparently there is another Defendant, one Emmanuel Ozoeneh, who may have been "either the sole inventor or co-inventor of the invention." It is clear, however, that Lawyers Mutual Liability Insurance Company of North Carolina issued a Professional Liability Insurance Policy to the Defendant Law Firm and a separate such Policy to one of the individual Defendants (not Mr. Ozoeneh). Carter v. Ozoeneh, 2010 WL 2889582 at *1.
Mr. Ozoeneh had a difficult time obtaining defense counsel. Lawyers Mutual undertook to pay Mr. Ozoeneh's legal fees "through summary judgment". Id. When Plaintiffs found out about this arrangement, they filed an Amended Complaint "asserting state law claims against Lawyers Mutual for tortious interference with contract, maintenance, and unfair and deceptive trade practices. Plaintiffs also allege bad faith, malice, and officious intermeddling by Lawyers Mutual." Id. at *2.
Thereafter, the Plaintiffs propounded Interrogatories and Requests for Production, the nature of which is not entirely clear from the Magistrate Judge's Opinion, and the Defendant Lawyers Mutual responded and produced privilege logs as to certain withheld materials, the identity or description of which is not entirely clear, either. However, it is clear that at least some of the withheld materials were delivered to the Magistrate Judge for an In Camera Inspection. Id. at *2-*3. The Magistrate Judge held that all of the materials inspected In Camera are privileged from discovery and do not need to be made available to the Plaintiffs. Some of the items are protected by the Attorney-Client Privilege, and two items -- one an attached Memorandum and the other an Expert Witness's Invoice -- are protected from discovery as Work Product, the Court held. Id. at *3.
The results in decided cases with reported In Camera Inspections of contested discovery materials are addressed by Dennis J. Wall, "Litigation and Prevention of Insurer Bad Faith" throughout Chapter 8, "Discovery From Insurers Concerning Their Handling of Third-Party Claims," and Chapter 12 concerning "Discovery From Insurers" in First Party Cases (Shepard's/McGraw-Hill First Edition, West Publishing Company Second Edition and 2010 Supplement).
Posted by Dennis J. Wall, Attorney At Law, a P.A. on August 03, 2010 at 05:20 AM in Attorney's Fees, Attorney-Client Privilege, Contract , Declaratory Judgment Actions, Discovery, Experts in Insurance Cases, Professional Liabilty | Permalink | Comments (0) | TrackBack (0)
Standard Lawyer Malpractice Policy Does Not Cover "Excessive MedMal Fees," ...
... Illinois Courts Hold.
A lawyer was sued in Illinois on "allegations that counsel retained an excessive amount of attorney fees from the settlement proceeds of a medical malpractice action." Download Continental Casualty Co. v. Donald T. Bertucci, Ltd. (Ill. 1st DCA Opinion Filed March 19, 2010), also published as Continental Casualty Co. v. Donald T. Bertucci, Ltd., 2010 WL 1033448 (Ill. 1st DCA March 19, 2010). The lawyer was also "named in attorney disciplinary proceedings" based on the same event. In Continental Casualty, the Illinois Courts applied standard legal professional errors and omissions Coverage provisions and held that there was no Coverage.
In the State Court action, the lawyer was sued on "claims of breach of contract, unjust enrichment, conversion, breach of fiduciary duty, fraud, and violation of the Illinois statute which limits contingent legal fees in medical malpractice actions". Id. (page numbers unavailable at time of publication). Both the Illinois Circuit or Trial Court, and the Illinois Appellate Court, held that there was no duty to defend or indemnify under the lawyer's policy. In basic and simple terms, the Illinois Courts compared the Policy Definition of covered "Damages" with the damages actually sought against the lawyer in the underlying lawsuit. The Courts concluded that there were no covered "Damages" at issue there. The former client's State Court lawsuit "alleges only noncovered, direct and consequential injuries from the excessive legal fees [the lawyer-Policyholder] charged against her assets; it does not allege 'damages' within the meaning of the policy."
In the Disciplinary Proceeding, the Illinois Trial Court had ruled that the event in question qualified as covered "legal services" covered under the Policy's "supplementary payments provision" for disciplinary proceedings. The Illinois Appellate Court reversed. There were no covered "legal services" at issue in the Disciplinary Proceeding, the Court held, any more than there were covered "legal services" at issue in the underlying State Court lawsuit, in the Appellate Court's view.
Fee disputes by any other name will be viewed by some Courts at least, as not being covered under standard Lawyer Professional Liability Insurance Policies.
Posted by Dennis J. Wall, Attorney At Law, a P.A. on April 12, 2010 at 06:03 AM in Damages, Professional Liabilty | Permalink | Comments (0) | TrackBack (0)


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Wrongful Death claims are never easy. No one should ever lose a loved one because of an accident that is caused by another's negligence. There is always a trail of facts or reports that show negligence on the part of the business or individual who is responsible for your loved one's death. We hold these individuals and businesses accountable for the terrible and tragic mistakes they've caused.
Gary Burger and Burger Law are a team of wrongful death lawyers who fight for the rights and recoveries of Carlinville families who have lost loved ones to reckless and negligent individuals. We don't accept anything less than the maximum amount recoverable under the law, and we will always place the needs of you and your family first. We don't charge any fees unless we win your family's claim. Call us at your earliest convenience at 1-314-542-2222, or use this form HERE.
These laws allow your family to seek recovery, compensation, and justice against those who caused the passing of your family member. It is important to collect all evidence regarding the claim as quickly as possible. Our Lawyers in Carlinville know how to get evidence from the defendants, how to depose the defendant, and how to guarantee that those responsible for the passing of your family member are fully punished for their negligent and reckless actions. If you have questions regarding how your family should start its wrongful death claim, how you can protect your claim, or how a personal injury lawyer can assist your family, please call us at 1-314-542-2222, or use this form HERE. We can help your family get the justice, compensation, and recovery it deserves.
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text:
\section{Introduction}
\label{sec 1}

  Evaluating asymptotically moments of $L$-functions at the central point is an important subject in analytical number theory as these moments have many arithmetic applications.  In this paper, we are interested in the families of twisted modular $L$-functions to a fixed modulus. For simplicity, we fix a holomorphic Hecke eigenform $f$ of weight $\kappa $ of level $1$. The Fourier expansion of
  $f$ at infinity can be written as
\[
f(z) = \sum_{n=1}^{\infty} \lambda (n) n^{\frac{\kappa -1}{2}} e(nz),
\]
where we denote $e(z) $ for $e^{2 \pi i z}$. Here the coefficients $\lambda (n)$ are real and satisfy $\lambda (1) =1$ and $|\lambda (n)|
\leq d(n)$ for $n \geq 1$ with $d(n)$ being the divisor function of $n$. We fix a modulus $q$ and assume that $q \not \equiv 2 \pmod 4$ throughout so that primitive Dirichlet characters modulo $q$ exist and denote $\chi$ for a primitive Dirichlet character modulo $q$.
For $\Re(s) > 1$, we define the twisted modular $L$-function $L(s, f \otimes \chi)$ by
\begin{align*}
L(s, f \otimes \chi) &= \sum_{n=1}^{\infty} \frac{\lambda (n)\chi(n)}{n^s}
 = \prod_{p} \left(1 - \frac{\lambda (p) \chi (p)}{p^s}  + \frac{\chi(p^2)}{p^{2s}}\right)^{-1}.
\end{align*}

  In \cite{Stefanicki}, T. Stefanicki obtained asymptotic formulas for the first and second moment of $L(\half, f \otimes \chi)$ over all primitive characters. The result given in \cite{Stefanicki} concerning the second moment is only valid for a restricted values of $q$ and this was later improved by P. Gao, R. Khan and G. Ricotta \cite{GKR} to hold for almost all $q$. Subsequent work with improvements on the error term can be found in \cite{BM15, BFKMM, KMS17}.

  In \cite{BM15}, V. Blomer and D. Mili\'{c}evi\'{c} not only studied the second moments of various modular $L$-functions
  in a much broader sense but also obtained sharp lower bounds for the second moment of mixed product of twisted modular $L$-functions.
  It is mentioned in \cite{BM15} that the method carries over to treat the $2k$-th moment for any integer $k$ as well
  and this was later achieved by G. Chen and X. He \cite{CH}.

  The aim of this paper is to establish sharp lower and upper bounds for the $2k$-th moments of central values of twisted modular $L$-functions to a fixed prime power modulus for real $k$. Throughout the paper, we denote $\phis(q)$ for the number of primitive characters  modulo $q$ and $\sumstar$ for the sum over primitive Dirichlet characters
  modulo $q$.  For lower bounds, we have the following result.
\begin{theorem}
\label{thmlowerbound}
    With notations as above. Let $q=q_0^{\nu}$ where $q_0$ is a large prime number and $\nu \geq 1$. For any real number $k \geq 0$, we have
\begin{align}
\label{lowerbound}
   \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2}, f \otimes \chi)|^{2k} \gg_k \phis(q)(\log q)^{k^2}.
\end{align}
\end{theorem}

   For upper bounds, our result is as follows.
\begin{theorem}
\label{thmupperbound}
    With notations as above. Let $q=q_0^{\nu}$ where $q_0$ is a large prime number and $\nu \geq 1$. For any real number $k$ such that $0 \leq k \leq 1$, we have
\begin{align}
\label{upperbound}
   \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},f \otimes \chi)|^{2k} \ll_k \phis(q)(\log q)^{k^2}.
\end{align}
\end{theorem}

   Our strategy of proofs for the above results largely follows from the lower bounds principle of W. Heap and K. Soundararajan \cite{H&Sound} as well as the upper bounds principle of M. Radziwi{\l\l} and K. Soundararajan \cite{Radziwill&Sound} concerning moments of general families of $L$-functions. These principles require one to be able to evaluate twisted moments. For our case, we shall employ the method in \cite{Stefanicki} to treat the twisted first moment and apply the tools developed in \cite{BM15} to treat the twisted second moment. Especially, an estimation for certain off-diagonal terms in \cite[Section 12]{BM15} plays a crucial role in our treatment for the twisted second moment.

  Combining Theorem \ref{thmlowerbound} and \ref{thmupperbound}, we have the following result concerning the order of magnitude of our family of $L$-functions.
\begin{theorem}
\label{thmorderofmag}
   With notations as above. Let $q=q_0^{\nu}$ where $q_0$ is a large prime number and $\nu \geq 1$. For any real number $k$ such that $0 \leq k \leq 1$, we have
\begin{align*}
   \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},\chi)|^{2k} \asymp \phis(q)(\log q)^{k^2}.
\end{align*}
\end{theorem}

   Our results above are in agreement with the conjectured formulas for these moments following from the general philosophy in the work of J. P. Keating and N. C. Snaith \cite{Keating&Snaith2000}.

\section{Preliminaries}
\label{sec 2}

  We reserve the letter $p$ for a prime number in this paper and
we note the following result concerning sums over primes.
\begin{lemma}
\label{RS} With notations as above and let $x \geq 2$.  We have for some constant $b$,
\begin{align}
\label{M1}
\sum_{p\le x} \frac{\lambda^2(p)}{p} = \log \log x + b+ O\Big(\frac{1}{\log x}\Big).
\end{align}
 Also,
\begin{align}
\label{M2}
\sum_{p\le x} \frac {\log p}{p} = \log x + O(1).
\end{align}
\end{lemma}

 The first assertion of Lemma \ref{RS} follows from the
Rankin-Selberg theory for $L(s, f)$,  which can be found in  \cite[Chapter 5]{iwakow}. The second assertion of Lemma \ref{RS}
is given in \cite[Lemma 2.7]{MVa1}.

  Next, we note the following approximate functional equations for $ L(\half, f \otimes \chi)$ and $|L(1/2, f \otimes \chi)|^2$.
\begin{lemma}
\label{PropDirpoly}
   For $X>0$, we have
\begin{align}
\label{lstapprox}
 L(\half, f \otimes \chi) = \sum^{\infty}_{n=1} \frac{\lambda(n)\chi(n)}{\sqrt{n}} W\left(\frac {nX}{q}\right)+\iota_{\chi}\sum^{\infty}_{n=1} \frac{\lambda(n)\overline \chi(n)}{\sqrt{n}} W\left(\frac {n}{qX}\right),
\end{align}
  where $\iota_{\chi}=i^{\kappa}\tau(\chi)^2/q$, $\tau(\chi)$ is the Gauss sum associated to $\chi$ and

$$ W(x) = \frac{1}{2\pi i} \int\limits_{(c)} \frac{\Gamma (\frac{\kappa}{2} +s)}{\Gamma (\frac{\kappa}{2})} e^{s^2} (2\pi x)^{-s} \> \frac{ds}{s}.$$
  We also have
\begin{align}
\label{lsquareapprox}
|L(\half, f \otimes \chi)|^2 = 2 \sum^{\infty}_{a, b=1} \frac{\chi(a) \overline{\chi}(b)}{\sqrt{ab}} \Wf \left(\frac {ab}{q^2}\right),
\end{align}
  where
$$ \Wf (x) = \frac{1}{2\pi i} \int\limits_{(c)} \frac{\Gamma\left(\frac{\kappa}{2} + s \right)^2}{(2\pi)^s \Gamma\left(\frac{\kappa}{2} \right)^2 } x^{-s} \> \frac{ds}{s}.$$
  Moreover, the functions $w(x), \Wf (x)$ are real valued and satisfy the bound that for any $c>0$,
\begin{align}
\label{W}
 W(x), \Wf (x)  \ll_c \min( 1 , x^{-c}).
\end{align}
\end{lemma}

    The functional equation given in \eqref{lstapprox} can be derived using standard arguments as those in the proof of \cite[Theorem 5.3]{iwakow}. The functional equation given in \eqref{lsquareapprox} can be found in \cite[Lemma 2.2]{GKR}.

\section{Outline of the Proofs}
\label{sec 2'}

   As the case $k=0$ is trivial and the case $k=1$ for
  \eqref{lowerbound}-\eqref{upperbound} is known from \cite{BM15}, we may assume in our proofs that $0< k \neq 1$ is a fixed positive real number.
We let $N, M$ be two large natural numbers depending on $k$ only and define a sequence of even natural
  numbers $\{ \ell_j \}_{1 \leq j \leq R}$ such that $\ell_1= 2\lceil N \log \log q\rceil$ and $\ell_{j+1} = 2 \lceil N \log \ell_j \rceil$ for
  $j \geq 1$, where we set $R$ to be the largest natural number satisfying $\ell_R >10^M$.  We shall choose $M$ large enough so that we have $\ell_{j} >
  \ell_{j+1}^2$ for all $1 \leq j \leq R-1$ and we note that this implies that
\begin{align}
\label{sumoverell}
  R \ll \log \log \ell_1, \quad \sum^R_{j=1}\frac 1{\ell_j} \leq \frac 2{\ell_R}.
\end{align}

    We define ${ P}_1$ to be the set of odd primes not exceeding $q^{1/\ell_1^2}$ and
${P_j}$ to be the set of primes lying in the interval $(q^{1/\ell_{j-1}^2}, q^{1/\ell_j^2}]$ for $2\le j\le R$. For each $1 \leq j \leq R$, we
write
\begin{equation*}
{\mathcal P}_j(\chi) = \sum_{p\in P_j} \frac{1}{\sqrt{p}} \chi(p), \quad  {\mathcal Q}_j(\chi, k) =\Big (\frac{c_k {\mathcal
P}_j(\chi) }{\ell_j}\Big)^{r_k\ell_j},
\end{equation*}
  where we define
\begin{align}
\label{r_k}
\begin{split}
  c_k= & 64 \max (1, k), \\
r_k =& \left\{
 \begin{array}
  [c]{ll}
  \lceil 1+1/k \rceil+1 & k>1,\\
  \lceil k /(2k-1) \rceil+1 & k<1.
 \end{array}
 \right.
 \end{split}
\end{align}
  We also define ${\mathcal Q}_{R+1}(\chi, k)=1$.

  For any non-negative integer $\ell$ and any real number $x$, we denote
\begin{equation*}
E_{\ell}(x) = \sum_{j=0}^{\ell} \frac{x^{j}}{j!}.
\end{equation*}
  We further define for each $1 \leq j \leq R$ and any real number $\alpha$,
\begin{align*}
{\mathcal N}_j(\chi, \alpha) = E_{\ell_j} (\alpha {\mathcal P}_j(\chi)), \quad \mathcal{N}(\chi, \alpha) = \prod_{j=1}^{R} {\mathcal
N}_j(\chi,\alpha).
\end{align*}

 We point out here that in the rest of the paper, when we use
 $\ll$ or the $O$-symbol to estimate various quantities presented, the implicit constants involved depend on $k$ only and are uniform with
 respect to $p, \chi$. We shall also make the convention that an empty product is defined to be $1$.

  The following two lemmas present in our setting the lower bounds principle of W. Heap and K. Soundararajan in \cite{H&Sound} and the
  upper bounds principle of M. Radziwi{\l\l} and K. Soundararajan in \cite{Radziwill&Sound}, respectively.

  The first one corresponds to the lower bounds principle.
\begin{lemma}
\label{lem1}
 With notations as above. For $0<k<1$, we have
\begin{align}
\label{basiclowerbound}
\begin{split}
\sumstar_{\substack{ \chi \shortmod q }}L(\tfrac{1}{2},f \otimes \chi)  \mathcal{N}(\chi, k-1) \mathcal{N}(\overline{\chi}, k)
 \ll & \Big ( \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2}, f \otimes \chi)|^{2k} \Big )^{1/2}\Big ( \sumstar_{\substack{ \chi \shortmod q
 }}|L(\tfrac{1}{2}, f \otimes  \chi)|^2 |\mathcal{N}(\chi, k-1)|^2  \Big)^{(1-k)/2} \\
 & \times \Big ( \sumstar_{\substack{ \chi \shortmod q }}   \prod^R_{j=1}\big ( |{\mathcal N}_j(\chi, k)|^2+ |{\mathcal Q}_j(\chi,k)|^2 \big )
 \Big)^{k/2}.
\end{split}
\end{align}
 For $k>1$, we have
\begin{align}
\label{basicboundkbig}
\begin{split}
 & \sumstar_{\substack{ \chi \shortmod q }}L(\tfrac{1}{2}, f \otimes \chi)  \mathcal{N}(\chi, k-1) \mathcal{N}(\overline{\chi}, k)
 \ll   \Big ( \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},f \otimes  \chi)|^{2k} \Big )^{\frac {1}{2k}}\Big ( \sumstar_{\substack{ \chi
 \shortmod q }} \prod^R_{j=1} \big ( |{\mathcal N}_j(\chi, k)|^2+ |{\mathcal Q}_j(\chi,k)|^2 \big ) \Big)^{\frac {2k-1}{2k}}.
\end{split}
\end{align}
  The implied constants in \eqref{basiclowerbound} and \eqref{basicboundkbig} depend on $k$ only.
\end{lemma}
\begin{proof}
   We consider the case $0<k<1$ first and we apply H\"older's inequality to see that the left side of \eqref{basiclowerbound} is
\begin{align*}
\begin{split}
 \leq & \Big ( \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},f \otimes  \chi)|^{2k} \Big )^{1/2}\Big ( \sumstar_{\substack{ \chi \shortmod q
 }}|L(\tfrac{1}{2},f \otimes \chi) \mathcal{N}(\chi, k-1)|^2   \Big)^{(1-k)/2}\Big ( \sumstar_{\substack{ \chi \shortmod q }} |\mathcal{N}(\chi,
 k)|^{2/k}|\mathcal{N}(\chi, k-1)|^{2}  \Big)^{k/2}.
\end{split}
\end{align*}

    As in the proof of \cite[Lemma 3.4]{Gao2021-3}, we have for $|z| \le aK/20$ with $0<a \leq 2$,
\begin{align}
\label{Ebound}
\Big| \sum_{r=0}^K \frac{z^r}{r!} - e^z \Big| \le \frac{|z|^{K}}{K!} \le \Big(\frac{a e}{20}\Big)^{K}.
\end{align}
  Setting $z=\alpha {\mathcal P}_j(\chi), K=\ell_j $ and $a=\min (|\alpha|, 2 )$ in \eqref{Ebound} then implies that when
  $|{\mathcal P}_j(\chi)| \le \ell_j/(20(1+|\alpha|))$,
\begin{align*}
{\mathcal N}_j(\chi, \alpha) \leq & \exp ( \alpha{\mathcal P}_j(\chi) )\left( 1+ \exp ( |\alpha {\mathcal P}_j(\chi)| ) \left( \frac{a e}{20} \right)^{ \ell_j}  \right) \leq \exp ( \alpha {\mathcal P}_j(\chi)  ) \left( 1+   e^{-\ell_j}  \right).
\end{align*}

  Similarly, we have
\begin{align*}
{\mathcal N}_j(\chi, \alpha) \geq & \exp ( \alpha{\mathcal P}_j(\chi) )\left( 1- \exp ( |\alpha {\mathcal P}_j(\chi)| ) \left( \frac{a e}{20} \right)^{ \ell_j} \right) \geq \exp ( \alpha {\mathcal P}_j(\chi)  ) \left( 1-   e^{-\ell_j}  \right).
\end{align*}

 We apply the above estimations to ${\mathcal N}_j(\chi, k-1), {\mathcal N}_j(\chi, k)$ to see that when $0<k<1$ and $|{\mathcal P}_j(\chi)| \le
 \ell_j/60$, then
\begin{align*}
\begin{split}
|\mathcal{N}_j(\chi,
 k-1)|^{2}
\leq & \exp ( 2(k-1) \Re {\mathcal P}_j(\chi)  ) \left( 1+  e^{-\ell_j}  \right)^{2} \\
 \leq & |{\mathcal N}_j(\chi, k)|^{\frac {2(k-1)}{k}} \left( 1+e^{-\ell_j} \right )^{2}\left( 1-e^{-\ell_j} \right )^{-\frac {2(k-1)}{k}}.
\end{split}
\end{align*}

 We then conclude that when $|{\mathcal P}_j(\chi)| \le \ell_j/60$,
\begin{align}
\label{est1}
|{\mathcal N}_j(\chi, k)^{\frac {1}{k}} {\mathcal N}_j(\chi, k-1)|^{2}
\leq & |{\mathcal N}_j(\chi, k)|^2 \left( 1+e^{-\ell_j} \right )^{2}\left( 1-e^{-\ell_j} \right )^{-\frac {2(k-1)}{k}}.
\end{align}

  On the other hand, we notice that when $|{\mathcal P}_j(\chi)| > \ell_j/60$,
\begin{align*}
\begin{split}
\max \Big (|{\mathcal N}_j(\chi, k-1)|, |{\mathcal N}_j(\chi, k)| \Big ) &\le \sum_{r=0}^{\ell_j} \frac{|{\mathcal P}_j(\chi)|^r}{r!} \le
|{\mathcal P}_j(\chi)|^{\ell_j} \sum_{r=0}^{\ell_j} \Big( \frac{60}{\ell_j}\Big)^{\ell_j-r} \frac{1}{r!}   \le \Big( \frac{64 |{\mathcal
P}_j(\chi)|}{\ell_j}\Big)^{\ell_j} .
\end{split}
\end{align*}
  It follows that when $|{\mathcal P}_j(\chi)| > \ell_j/60$, we have
\begin{align}
\label{est2}
|{\mathcal N}_j(\chi, k)^{\frac {1}{k}} {\mathcal N}_j(\chi, k-1)|^{2}
& \leq \Big( \frac{64 |{\mathcal P}_j(\chi)|}{\ell_j}\Big)^{2(1+1/k)\ell_j} \leq  |{\mathcal Q}_j(\chi, k)|^2.
\end{align}

 We apply \eqref{est1} and \eqref{est2} to deduce that when $0<k<1$, we have
\begin{align*}
\begin{split}
 \sumstar_{\substack{ \chi \shortmod q }} |\mathcal{N}(\chi,
 k)|^{2/k}|\mathcal{N}(\chi, k-1)|^{2} \le &
\sumstar_{\substack{ \chi \shortmod q }}   \Big ( \prod^R_{j=1} \Big (|{\mathcal N}_j(\chi, k)|^2 \left( 1+e^{-\ell_j} \right )^{2}\left( 1-e^{-\ell_j} \right )^{-\frac {2(k-1)}{k}}+   |{\mathcal Q}_j(\chi, k)|^2  \Big )\Big
) \\
\leq & \prod^R_{j=1} \max \Big( \left( 1+e^{-\ell_j} \right )^{2}\left( 1-e^{-\ell_j} \right )^{-\frac {2(k-1)}{k}}, 1 \Big ) \sumstar_{\substack{ \chi \shortmod q }}   \prod^R_{j=1} \Big (|{\mathcal N}_j(\chi, k)|^2 +  |{\mathcal Q}_j(\chi, k)|^2  \Big )  \\
\ll & \sumstar_{\substack{ \chi \shortmod q }} \prod^R_{j=1} \Big ( |{\mathcal N}_j(\chi, k)|^2 +  |{\mathcal Q}_j(\chi, k)|^2   \Big ),
\end{split}
\end{align*}
 where the last estimation above follows by noting that
\begin{align*}
\begin{split}
 \prod^R_{j=1} \max \Big( \left( 1+e^{-\ell_j} \right )^{2}\left( 1-e^{-\ell_j} \right )^{-\frac {2(k-1)}{k}}, 1 \Big ) \ll 1.
\end{split}
\end{align*}

 Next, we consider the case $k>1$ and we apply H\"older's inequality again to see that the left side of \eqref{basicboundkbig} is
\begin{align}
\label{holderkbig}
\begin{split}
 \leq  \Big ( \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2}, f \otimes \chi)|^{2k} \Big )^{\frac {1}{2k}}\Big ( \sumstar_{\substack{ \chi
 \shortmod q }} |\mathcal{N}(\chi, k)\mathcal{N}(\chi, k-1)|^{\frac {2k}{2k-1}}  \Big)^{\frac {2k-1}{2k}}.
\end{split}
\end{align}

  We argue similar to above to see that when  $|{\mathcal P}_j(\chi)| \le \ell_j/(40k)$,
\begin{align}
\label{prodNkbig}
\begin{split}
 |\mathcal{N}_j(\chi, k)\mathcal{N}_j(\chi, k-1)|^{\frac {2k}{2k-1}} \leq |{\mathcal N}_j(\chi, k)|^2 \Big( 1+ e^{-\ell_j}  \Big)^{\frac {2k}{2k-1}}\left( 1-e^{-\ell_j} \right )^{-\frac {2(k-1)}{2k-1}}.
\end{split}
\end{align}
  Similarly, when $|{\mathcal P}_j(\chi)| > \ell_j/(40k)$, we have
\begin{align*}
\begin{split}
 |\mathcal{N}_j(\chi, k)\mathcal{N}_j(\chi, k-1)|^{\frac {2k}{2k-1}}
  \leq  |{\mathcal Q}_j(\chi, k)|^2.
\end{split}
\end{align*}
  Combining \eqref{holderkbig}, \eqref{prodNkbig} and arguing as above, we readily deduce the estimation given in \eqref{basicboundkbig}.
  This completes the proof of the lemma.
\end{proof}

  Our next lemma corresponds to the upper bounds principle.
\begin{lemma}
\label{lem2}
 With notations as above. We have for $0<k<1$,
\begin{align}
\label{basiclowerbound2}
\begin{split}
& \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},f \otimes \chi)|^{2k} \\
 \ll & \Big ( \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},f \otimes \chi)|^2 \sum^{R}_{v=0} \Big (\prod^v_{j=1} |\mathcal{N}_j(\chi, k-1)|^2 \Big ) |{\mathcal
 Q}_{v+1}(\chi, k)|^{2}
 \Big)^{k} \Big ( \sumstar_{\substack{ \chi \shortmod q }}  \sum^{R}_{v=0}  \Big (\prod^v_{j=1}|\mathcal{N}_j(\chi, k)|^2\Big )|{\mathcal
 Q}_{v+1}(\chi, k)|^{2} \Big)^{1-k},
\end{split}
\end{align}
  where the implied constants depend on $k$ only.
\end{lemma}
\begin{proof}
  Using arguments similar to those in the proof of Lemma \ref{lem1}, we see that when $|{\mathcal P}_j(\chi)| \le \ell_j/60$,
\begin{align}
\label{prodNlowerbound}
 |\mathcal{N}_j(\chi, k-1)|^{2k}|\mathcal{N}_j(\chi, k)|^{2(1-k)} \geq \big(1- e^{-\ell_j} \big )^2.
\end{align}

   If there exists an integer $0 \leq v \leq R-1$ such that $| \mathcal{P}_j (\chi) | \leq \ell_j/60$ for all $j \leq v$ and that $|
  \mathcal{P}_{v+1} (\chi) | > \ell_{v+1}/60$,  we deduce from the above and the observation that $|{\mathcal Q}_{v+1}(\chi, k)| \geq 1$ when
  $|{\mathcal P}_{v+1}(\chi)| \ge \ell_{v+1}/60$ that
\begin{align*}
 \Big ( \prod^v_{j=1}|\mathcal{N}_j(\chi, k-1)|^{2k}|\mathcal{N}_j(\chi, k)|^{2(1-k)} \Big )|{\mathcal Q}_{v+1}(\chi, k)|^2 \gg 1.
\end{align*}

  If no such $v$ exists, then $| \mathcal{P}_j (\chi) | \leq \ell_j/60$ for all $1 \leq j \leq R$. Thus the estimation
  \eqref{prodNlowerbound} holds for all $j$ and we have
\begin{align*}
 \prod^R_{j=1}|\mathcal{N}_j(\chi, k-1)|^{2k}|\mathcal{N}_j(\chi, k)|^{2(1-k)} \gg 1.
\end{align*}

  In either case, we conclude that
\begin{align*}
 \Big (\sum^R_{v=0}\Big ( \prod^v_{j=1}|\mathcal{N}_j(\chi, k-1)|^2 \Big )|{\mathcal Q}_{v+1}(\chi, k)|^2 \Big )^{k}\Big (\sum^R_{v=0}\Big ( \prod^v_{j=1}|\mathcal{N}_j(\chi, k)|^2\Big ) |{\mathcal Q}_{v+1}(\chi, k)|^2 \Big )^{1-k}  \gg 1.
\end{align*}
   It follows from this that we have
\begin{align*}
 & \sumstar_{\substack{ \chi \shortmod q }} |L(\half, f \otimes \chi)|^{2k} \\
 \ll & \sumstar_{\substack{ \chi \shortmod q }} |L(\half, f \otimes \chi)|^{2k} \Big (\sum^R_{v=0}\Big ( \prod^v_{j=1}|\mathcal{N}_j(\chi, k-1)|^2 \Big )|{\mathcal Q}_{v+1}(\chi, k)|^2 \Big )^{k}
 \times \Big (\sum^R_{v=0}\Big (  \prod^v_{j=1}|\mathcal{N}_j(\chi, k)|^2\Big ) |{\mathcal Q}_{v+1}(\chi, k)|^2 \Big )^{1-k}.
\end{align*}
     Applying H\"older's inequality to the last expression above leads to the estimation given in \eqref{basiclowerbound2} and this completes the proof of the lemma.
\end{proof}

  We apply Lemma \ref{lem1} and Lemma \ref{lem2} to see that in order to prove Theorem \ref{thmlowerbound} and Theorem
   \ref{thmupperbound}, it suffices to establish the following three propositions.
\begin{proposition}
\label{Prop4} With notations as above. We have for $k>0$,
\begin{align*}
\sumstar_{\substack{ \chi \shortmod q }}L(\tfrac{1}{2}, f \otimes \chi) \mathcal{N}(\overline{\chi}, k) \mathcal{N}(\chi, k-1) \gg \phis(q)(\log q)^{ k^2
} .
\end{align*}
\end{proposition}

\begin{proposition}
\label{Prop5} With notations as above. We have for $0<k < 1$,
\begin{align*}
\max \Big (  \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},f \otimes \chi)\mathcal{N}(\chi, k-1)|^2, \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},f \otimes \chi)|^2 \sum^{R}_{v=0}\Big (\prod^v_{j=1}|\mathcal{N}_j(\chi, k-1)|^{2}\Big ) |{\mathcal
 Q}_{v+1}(\chi, k)|^2 \Big )   \ll
\phis(q)(\log q)^{ k^2 }.
\end{align*}
\end{proposition}

\begin{proposition}
\label{Prop6} With notations as above. We have for $k>0$,
\begin{align*}
\max \Big ( \sumstar_{\substack{ \chi \shortmod q }}\prod^R_{j=1}\big ( |{\mathcal N}_j(\chi, k)|^2+ |{\mathcal Q}_j(\chi,k)|^2 \big ),  \sumstar_{\substack{ \chi \shortmod q }} \sum^{R}_{v=0} \Big ( \prod^v_{j=1}|\mathcal{N}_j(\chi, k)|^{2}\Big )|{\mathcal
 Q}_{v+1}(\chi, k)|^2 \Big )   \ll \phis(q)(\log q)^{ k^2 }.
\end{align*}
\end{proposition}

   We shall prove the above propositions in the rest of the paper.

\section{Proof of Proposition \ref{Prop4}}
\label{sec 4}

    We define $\widetilde{\lambda}(n)$ to be the completely multiplicative function $\widetilde{\lambda}(p)=\lambda(p)$ on primes $p$, we also
    define $w(n)$ to be the multiplicative function such that
    $w(p^{\alpha}) = \alpha!$ for prime powers $p^{\alpha}$.
    We further denote $\Omega(n)$ for the number of prime powers dividing $n$ and let $b_j(n), 1 \leq j \leq R$ be functions such that $b_j(n)$ only takes
    values $0$ or $1$ and $b_j(n)=1$ if and only if $\Omega(n) \leq \ell_j$ and the primes dividing $n$ are all from the interval $P_j$.
    We use these notations to write ${\mathcal N}_j(\chi, \alpha)$ as
\begin{equation}
\label{5.1}
{\mathcal N}_j(\chi, \alpha) = \sum_{n_j} \frac{\widetilde{\lambda}(n_j)}{\sqrt{n_j}} \frac{\alpha^{\Omega(n_j)}}{w(n_j)}  b_j(n_j) \chi(n_j), \quad 1\le j\le R.
\end{equation}
    Note that each ${\mathcal N}_j(\chi, \alpha)$ is a short Dirichlet polynomial since $b_j(n_j)=0$ unless $n_j \leq
    (q^{1/\ell_j^2})^{\ell_j}=q^{1/\ell_j}$. We then deduce that both ${\mathcal N}(\chi, k)$ and ${\mathcal N}(\chi, k-1)$ are short Dirichlet
    polynomials whose lengths are both at most $q^{1/\ell_1+ \ldots +1/\ell_R} < q^{2/10^{M}}$ by \eqref{sumoverell}.
    Note further that we have $\widetilde{\lambda}_f(n_j) \leq 2^{\Omega(n_j)}$ since $\lambda_f(p) \leq 2$ and $b_j(n_j)$ restricts $n_j$
     to satisfy $\Omega(n_j) \leq \ell_j$. This observation allows us to also write ${\mathcal N}_j(\chi, \alpha)$ as
\begin{align}
\label{Nj}
  {\mathcal N}_j(\chi, \alpha)=\sum_{n_j \leq q^{1/\ell_j}}\frac{c_{n_j}b_j(n_j)}{\sqrt{n_j}}\chi(n_j),
\end{align}
   where we have, for some constant $B_0(\alpha)$ depending on $\alpha$ only,
\begin{align*}
  |c_{n_j}| \leq B_0(\alpha)^{\ell_j}.
\end{align*}

  We apply \eqref{Nj} to write for simplicity that
\begin{align*}
 {\mathcal N}(\chi, k-1)= \sum_{a  \leq q^{2/10^{M}}} \frac{x_a}{\sqrt{a}} \chi(a), \quad \mathcal{N}(\overline{\chi}, k) = \sum_{b  \leq
 q^{2/10^{M}}} \frac{y_b}{\sqrt{b}}\overline{\chi}(b),
\end{align*}
    where for some constant $B(k)$ depending on $k$ only, we have
\begin{align}
\label{xybounds}
 x_a, y_b \ll B(k)^{\sum^R_{j=1}\ell_j}  \ll B(k)^{R\ell_1} \ll q^{\varepsilon}.
\end{align}
  The last estimation above follows from \eqref{sumoverell}.

  We now deduce from Lemma \ref{PropDirpoly} that
\begin{align}
\label{Lfirstmoment}
\begin{split}
& \sumstar_{\substack{ \chi \shortmod q }}L(\tfrac{1}{2},f \otimes \chi) \mathcal{N}(\overline{\chi}, k) \mathcal{N}(\chi, k-1) \\
= & \sumstar_{\substack{ \chi \shortmod q }}\sum_{m}\frac {\lambda(m)\chi(m)}{\sqrt{m}}\mathcal{N}(\overline{\chi}, k) \mathcal{N}(\chi,
k-1) W\left(\frac {mX}{q}\right)+\sumstar_{\substack{ \chi \shortmod q }}\iota_{\chi} \sum_{m}\frac {\lambda(m)\overline\chi(m)}{\sqrt{m}}\mathcal{N}(\overline{\chi}, k) \mathcal{N}(\chi,
k-1) W\left(\frac {m}{qX}\right).
\end{split}
\end{align}

  We denote $\mu$ for the M\"obius function and note the following relation (see \cite[(2.3)]{Stefanicki})
\begin{align}
\label{sumchistar}
 & \sumstar_{\substack{ \chi \shortmod q }}\chi(a)=\sum_{c | (q, a-1)}\mu(q/c)\phi(c).
\end{align}
   In particular, setting $a=1$ above implies that
\begin{align}
\label{chistar}
 & \phis(q)=\sum_{c | q}\mu(q/c)\phi(c).
\end{align}

  We apply \eqref{sumchistar} to see that the right side expression in \eqref{Lfirstmoment} equals to
\begin{align*}
 \sum_{c | q}\mu(q/c)\phi(c) \sum_{a} \sum_{b} \sum_{\substack{am \equiv b \bmod c}}\frac {\lambda(m) x_a y_b}{\sqrt{abm}}W\left(\frac {mX}{q}\right)+\sum_{a} \sum_{b} \sum_{\substack{m}}\frac {\lambda(m)x_ay_b}{\sqrt{abm}}W\left(\frac {m}{qX}\right)\sumstar_{\substack{ \chi \shortmod q }}\iota_{\chi}\chi(a)\overline \chi(mb).
\end{align*}

  We evaluate the last summation above by applying the definition of $\iota_{\chi}$ to see that
\begin{align*}
 \sumstar_{\substack{ \chi \shortmod q }}\iota_{\chi}\chi(a)\overline \chi(mb) = \frac {i^{\kappa}}{q}  \sumstar_{\substack{ \chi \shortmod q }} \chi(a)\overline \chi(mb)  \sumstar_{\substack{ v \shortmod q }} \chi(v) S(1,v,q)=\frac {i^{\kappa}}{q}\sum_{c | q}\mu(q/c)\phi(c)\sumstar_{\substack{ v \shortmod q \\ av \equiv mb \shortmod c} } S(1,v,q),
\end{align*}
  where $S$ is the Kloosterman's sum defined by
\begin{align*}
  S(u,v, q)=\sumstar_{\substack{ h \shortmod q }} \exp (\frac {uh+v\overline h}{q}).
\end{align*}

   It follows from the well-known Weil's bound for Kloosterman's sum (see \cite[Corollary 11.20]{iwakow}) that we have
\begin{align*}
  |S(1,v, q)| \leq d(q)q^{1/2}.
\end{align*}

   We then deduce from this that
\begin{align*}
 \sumstar_{\substack{ \chi \shortmod q }}\iota_{\chi}\chi(a)\overline \chi(mb) \leq \frac {1}{q}\sum_{c | q}\phi(c)\frac {q}{c} d(q)q^{1/2} \ll q^{\half+\varepsilon}.
\end{align*}

  The above then implies that we have
\begin{align}
\label{Firstmomentsum2}
& \sum_{a} \sum_{b} \sum_{\substack{m}}\frac {\lambda(m)x_ay_b}{\sqrt{abm}}W\left(\frac {m}{qX}\right)\sumstar_{\substack{ \chi \shortmod q }}\iota_{\chi}\chi(a)\overline \chi(mb)
 \ll  q^{4/10^{M}}q^{1/2+\varepsilon}X^{\varepsilon}\sqrt{qX}.
\end{align}

  It remains to evaluate
\begin{align}
\label{Firstmomentsum1}
 \sum_{c | q}\mu(q/c)\phi(c) \sum_{a} \sum_{b} \sum_{\substack{am \equiv b \bmod c}}\frac {\lambda(m) x_a y_b}{\sqrt{abm}}W\left(\frac {mX}{q}\right).
\end{align}

  We first consider the contribution from the terms $am=b+l c$ with $l \geq 1$ above (note that we may take $M$ large enough
  so that we can not have $b > am$ in our case). By the rapid decay of $W(x)$ given in \eqref{W}, we may assume that $m \leq (q/X)^{1+\varepsilon}$. This then implies that $l \leq  q^{1+2/10^{M}+\varepsilon}/(Xc)$, so that we deduce together with the observation that $x_a, y_b \ll q^{\varepsilon}$ that the total
  contribution from these terms is
\begin{align}
\label{Firstmomentsum1nondiag1}
 \ll &  \sum_{c | q} \phi(c)  q^{\varepsilon}X^{\varepsilon} \sum_{b  \leq q^{2/10^{M}}}  \sum_{l \leq q^{1+2/10^{M}+\varepsilon}/(Xc)}\frac {d(b+l c)}{\sqrt{bdl}} \ll X^{-1/2+\varepsilon}q^{\half+2/10^{M}+\varepsilon}.
\end{align}

  We now set $X=q^{-16/10^{M}}$ to see from \eqref{Lfirstmoment}, \eqref{chistar}-\eqref{Firstmomentsum1nondiag1} that we have
\begin{align*}
& \sumstar_{\substack{ \chi \shortmod q }}L(\tfrac{1}{2},f \otimes \chi) \mathcal{N}(\overline{\chi}, k) \mathcal{N}(\chi, k-1)
\gg \phis(q) \sum_{a} \sum_{b} \sum_{\substack{m \leq (q/X)^{1+\varepsilon} \\ am = b }}\frac {\lambda(m) x_a y_b}{\sqrt{abm}}
= \phis(q) \sum_{b} \frac {y_b}{b} \sum_{\substack{a, m \\ am = b }}\lambda(m)x_a,
\end{align*}
  where the last equality above follows from the observation that $b \leq q^{2/10^{M}}<(q/X)^{1+\varepsilon}$.

   Notice that
\begin{align}
\label{sumab}
\begin{split}
\sum_{b} \frac {y_b}{b} \sum_{\substack{a | b }}\lambda(m)x_a=& \prod^R_{j=1}\Big ( \sum_{n_j} \frac{\widetilde{\lambda}(n_j)}{n_j} \frac{k^{\Omega(n_j)}}{w(n_j)}
b_j(n_j)\sum_{n'_j|n_j} \frac{\widetilde{\lambda}(n'_j)(k-1)^{\Omega(n'_j)}}{w(n'_j)}  b_j(n'_j)\lambda(n_j/n'_j) \Big ) \\
=& \prod^R_{j=1}\Big ( \sum_{n_j} \frac{\widetilde{\lambda}(n_j)}{n_j} \frac{k^{\Omega(n_j)}}{w(n_j)}  b_j(n_j)\sum_{n'_j|n_j}  \frac{\widetilde{\lambda}(n'_j)(k-1)^{\Omega(n'_j)}}{w(n'_j)}\lambda(n_j/n'_j)
\Big ),
\end{split}
\end{align}
  where the last equality above follows by noting that $b_j(n_j)=1$ implies that $b_j(n'_j)=1$ for all $n'_j|n_j$.

  We consider the sum above over $n_j$ for a fixed $1 \leq j \leq R$ in \eqref{sumab}. Note that the factor $b_j(n_j)$ restricts $n_j$ to have
  all prime factors in $P_j$ such that $\Omega(n_j) \leq \ell_j$. If we remove the restriction on $\Omega(n_j)$, then the sum becomes
\begin{align}
\label{6.02}
\begin{split}
& \prod_{\substack{p\in P_j }} \Big( \sum_{i=0}^{\infty} \frac{\lambda^i(p)}{p^i} \frac{k^{i}}{i!}\Big ( \sum_{l=0}^{i}
\frac{(k-1)^{l}}{l!} \lambda^l(p)\lambda(p^{i-l})\Big ) \Big)
= \prod_{\substack{p\in P_j }}\Big (1+ \frac {k^2\lambda^2(p)}p+O(\frac 1{p^2}) \Big ).
\end{split}
\end{align}

   On the other hand, using Rankin's trick by noticing that $2^{\Omega(n_j)-\ell_j}\ge 1$ if $\Omega(n_j) > \ell_j$,  we see that the error
   introduced this way does not exceed
\begin{align*}
\begin{split}
 & \sum_{n_j} \frac{\widetilde{\lambda}(n_j)}{n_j} \frac{k^{\Omega(n_j)}}{w(n_j)}2^{\Omega(n_j)-\ell_j}  \sum_{n'_j|n_j}  \frac{\widetilde{\lambda}(n'_j)|1-k|^{\Omega(n'_j)}}{w(n'_j)}
  \\
\le & 2^{-\ell_j} \prod_{\substack{p\in P_j }} \Big( \sum_{i=0}^{\infty} \frac{\lambda^i(p)}{p^i} \frac{(2k)^{i}}{i!}\Big ( \sum_{l=0}^{i}
\frac{|1-k|^{l}}{l!}\lambda^l(p)\lambda(p^{i-l}) \Big ) \Big) \\
\le & 2^{-\ell_j} \prod_{\substack{p\in P_j }} \Big( 1+ \frac {2k(1+|1-k|)\lambda^2(p)}p+O(\frac 1{p^2})\Big) \\
\leq & 2^{-\ell_j/2}\prod_{\substack{p\in P_j }}\Big (1+ \frac {k^2\lambda^2(p)}p+O(\frac 1{p^2}) \Big ),
\end{split}
\end{align*}
  where the last estimation above follows by taking $N$ large enough and the
bound (which is a consequence of Lemma \ref{RS}) that
\begin{align}
\label{boundsforsumoverp}
   \frac 1{4N} \ell_j \leq \sum_{p \in P_j}\frac {\lambda^2(p)}{p} \leq \frac {2}N \ell_j.
\end{align}

  We then deduce from this, \eqref{6.02} and Lemma \ref{RS} that we have
\begin{align*}
& \sumstar_{\substack{ \chi \shortmod q }}L(\tfrac{1}{2},\chi) \mathcal{N}(\overline{\chi}, k) \mathcal{N}(\chi, k-1)
\gg \phis(q) \prod^R_{j=1}\Big (1+O( 2^{-\ell_j/2})\Big )\prod_{\substack{p\in P_j }}\Big (1+ \frac {k^2\lambda^2(p)}p+O(\frac 1{p^2}) \Big ) \gg
\phis(q)(\log q)^{k^2}.
\end{align*}
 This completes the proof of the proposition.

\section{Proof of Proposition \ref{Prop5}}
\label{sec 5}

  Observe that for a fixed integer $v$ such that $1 \leq v \leq R-1$, we have
\begin{align*}
\begin{split}
 & \sumstar_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},f \otimes \chi)|^2 \Big (\prod^v_{j=1}|\mathcal{N}_j(\chi, k-1)|^2 \Big )|{\mathcal
 Q}_{v+1}(\chi, k)|^{2}
 \leq  \sum_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},f \otimes \chi)|^2 \Big (\prod^v_{j=1}|\mathcal{N}_j(\chi, k-1)|^2 \Big )|{\mathcal
 Q}_{v+1}(\chi, k)|^{2} .
\end{split}
\end{align*}
    Since the sum over $e^{-\ell_j/2}$ converges, we deduce from the above that it remains to show that
\begin{align*}
\begin{split}
 \sum_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},f \otimes \chi)|^2 \Big (\prod^v_{j=1}|\mathcal{N}_j(\chi, k-1)|^2 \Big )|{\mathcal
 Q}_{v+1}(\chi, k)|^{2}
 \ll & \phis(q)e^{-\ell_{v+1}/2}(\log q)^{k^2}.
\end{split}
\end{align*}

 Recall the definition of $c_k, r_k$ given in \eqref{r_k}. We further define the function $p_{v+1}(n)$ such that $p_{v+1}(n)=0$ or $1$, and that $p_{v+1}(n)=1$ if and only if $\Omega(n)=r_k\ell_{v+1}$ and all the prime factors of $n$ are from the interval $P_{v+1}$. Using these together with the notations in Section \ref{sec 4}, we see that
\begin{align}
\label{Pexpression}
  {\mathcal Q}_{v+1}(\chi, k)^{r_k\ell_{v+1}} =&  \Big( \frac{c_k  }{\ell_{v+1}}\Big)^{r_k\ell_{v+1}}\sum_{ \substack{ n_{v+1}}} \frac{\widetilde{\lambda}(n_{v+1})}{\sqrt{n_{v+1}}}\frac{(c_k\ell_{v+1})!
  }{w(n_{v+1})}\chi(n_{v+1})p_{v+1}(n_{v+1}).
\end{align}

  Note that $\prod^v_{j=1}\mathcal{N}_j(\chi, k-1){\mathcal
 Q}_{v+1}(\chi, k)$ is a short Dirichlet polynomial whose length does not exceed
$$q^{1/\ell_1+ \ldots +1/\ell_v+r_k/\ell_{v+1}} < q^{2r_k/10^{M}}.$$
  We apply \eqref{5.1}, \eqref{Pexpression} and the above observation to write for simplicity that
$$ \Big (\prod^v_{j=1}|\mathcal{N}_j(\chi, k-1)|^2 \Big )|{\mathcal
 Q}_{v+1}(\chi, k)|^{2} = \Big( \frac{c_k  }{\ell_{v+1}}\Big)^{2r_k\ell_{v+1}}((r_k\ell_{v+1})!)^2 \sum_{a,b \leq q^{2r_k/10^{M}}} \frac{u_a u_b}{\sqrt{ab}}\chi(a)\overline{\chi}(b).$$

   Similar to \eqref{xybounds}, we note that for all $a, b$,
\begin{align}
\label{ubounds}
 u_a, u_b \ll q^{\varepsilon}.
\end{align}

  We apply \eqref{lsquareapprox} to see that
\begin{align}
\label{LNsquaresum}
\begin{split}
& {\sum_{\chi \shortmod q}} |L(\half, f \otimes \chi)|^2\Big (\prod^v_{j=1}|\mathcal{N}_j(\chi, k-1)|^2 \Big )|{\mathcal
 Q}_{v+1}(\chi, k)|^{2} \\
=& 2\Big( \frac{c_k  }{\ell_{v+1}}\Big)^{2r_k\ell_{v+1}}((r_k\ell_{v+1})!)^2 \sum_{a,b \leq q^{2r_k/10^{M}}} \frac{u_a u_b}{\sqrt{ab}} \sum_{m, n} \frac{\lambda(m)\lambda(n)}{\sqrt{mn}} \Wf \left(\frac {mn}{q^2}\right)
{\sum_{\chi}} \chi(ma) \overline{\chi}(nb) \\
=& \phi(q) \Big( \frac {c_k }{\ell_{v+1}}\Big)^{2r_k\ell_{v+1}}((r_k\ell_{v+1})!)^2 \sum_{a,b \leq q^{2r_k/10^{M}}} \frac{u_a u_b}{\sqrt{ab}} \sum_{\substack{m,n \\ (mn, q)=1 \\ ma \equiv nb \,{\rm mod} \, q}}
\frac{\lambda(m)\lambda(n)}{\sqrt{mn}} \Wf \left(\frac {mn}{q^2}\right).
\end{split}
\end{align}

  We now consider the contribution of the terms $ma \neq nb$ in the last expression of \eqref{LNsquaresum}. Due to the rapid decay of $\Wf(x)$ given in \eqref{W}, we may assume that $mn \leq q^{2+\varepsilon}$.  We apply \cite[Lemma 1.6]{BFKMM} to see that there exist two non-negative function ${\mathcal V}_1(x), {\mathcal V}_2(x)$ supported on $[1/2,2]$,  satisfying
\begin{align}
\label{Vbounds}
\begin{split}
 {\mathcal V}^{(j)}_i(x) \ll_{j, \varepsilon} q^{j\varepsilon}.
\end{split}
\end{align}
  Moreover, we have the following smooth partition of unity:
\begin{align*}
\begin{split}
  \sum_{k \geq 0}{\mathcal V}_i(\frac x{2^k})=1, \quad i=1,2.
\end{split}
\end{align*}

  We also note the estimations
\begin{align}
\label{Stirling}
  (\frac ne)^n \leq n! \leq n(\frac ne)^n.
\end{align}

  It follows from this and the definition of $\ell_{v+1}$ given in Section \ref{sec 2'} that we have
\begin{align}
\label{lv1est}
 \Big( \frac {c_k }{\ell_{v+1}}\Big)^{2r_k\ell_{v+1}}((r_k\ell_{v+1})!)^2  \leq (r_k\ell_{v+1})^2 \Big( \frac{c_k r_k }{e } \Big)^{2r_k\ell_{v+1}} \ll q^{\varepsilon}.
\end{align}

  Applying the above, the definition of $\Wf (x)$ given in Lemma \ref{PropDirpoly} and the estimations given in \eqref{ubounds}, we see that the terms $ma \neq nb$ in the last expression of \eqref{LNsquaresum} contributes
\begin{align}
\label{offdiagbounds}
\begin{split}
 \ll & \frac {q^{1+\varepsilon}}{\sqrt{AB}} \sum_{a,b \leq q^{2r_k/10^{M}}}\frac {1}{\sqrt{ab}} \sum_{\substack{A=2^{k_1}, B=2^{k_2} \\ k_1, k_2 \geq 0 \\ AB \leq q^{2+\varepsilon}}} \int\limits_{(\varepsilon)} \Big |
\frac{\Gamma\left(\frac{\kappa}{2} + s \right)^2}{(2\pi )^s \Gamma\left(\frac{\kappa}{2} \right)^2 }\sum_{\substack{ma \neq nb \\ (mn, q)=1 \\ ma \equiv nb \,{\rm mod} \, q}}
\lambda(m)\lambda(n)V_1 \left(\frac {m}{A}\right) V_2 \left(\frac {n}{B}\right)\big (\frac {q}{mn}\big )^{s} \Big |
\frac{|ds|}{|s|},
\end{split}
\end{align}
 where
\begin{align*}
  V_i \left(x \right)=x^{-\half-s}{\mathcal V}_i(x), \quad i=1,2.
\end{align*}
  Due to the rapid decay of $\Gamma(s)$ on the vertical line, we may truncate the integral in \eqref{offdiagbounds} to $\Im(s) \leq (\log 5q)^2$ with a negligible error. This implies that the bounds given in \eqref{Vbounds} are also satisfied by $V_i (i=1,2)$ and that the expression in \eqref{offdiagbounds} can be further bounded by
\begin{align*}
\begin{split}
 \sum_{\substack{A=2^{k_1}, B=2^{k_2} \\ k_1, k_2 \geq 0 \\ AB \leq q^{2+\varepsilon}}}E(A,B),
\end{split}
\end{align*}
  where $A, B \geq 1$, $AB\leq q^{2+\varepsilon}$ and that
\begin{align*}
\begin{split}
& E(A, B)=\frac {q^{1+\varepsilon}}{\sqrt{AB}} \sum_{a,b \leq q^{2r_k/10^{M}}}\frac {1}{\sqrt{ab}}  \Big |\sum_{\substack{ma \neq nb \\ (mn, q)=1 \\ ma \equiv nb \,{\rm mod} \, q}}
\lambda(m)\lambda(n) V_1 \left(\frac {m}{A}\right) V_2 \left(\frac {n}{B}\right) \Big |.
\end{split}
\end{align*}

  We are thus led to estimate $E(A, B)$ for integers $A, B \geq 1, AB \leq q^{2+\varepsilon}$ and functions $V_1, V_2$ satisfying \eqref{Vbounds}.
In fact, this work has already been done in \cite[Section 12]{BM15} and
it follows from the result given in the first display below \cite[(12.7)]{BM15} that we have upon setting $X=q^{2r_k/10^{M}}, \theta=0$ there that $E(A,B) \ll q^{1-\varepsilon}$. This implies that  the contribution of the terms $ma \neq nb$ in the last expression of \eqref{LNsquaresum} is negligible.

 It remains to consider the terms $ma = nb$ in the last expression of \eqref{LNsquaresum}. We write $m = \frac{\alpha b}{(a,b)}, n =
 \frac{\alpha a}{(a,b)}$ and apply the estimation in \eqref{lv1est} to see that these terms are
\begin{align}
\label{mainterm}
\ll & \phi(q)(r_k\ell_{v+1})^2 \Big( \frac{c_k r_k }{e } \Big)^{2r_k\ell_{v+1}} \sum_{a,b \leq q^{2r_k/10^{M}}} \frac{(a,b)}{ab} u_a u_b  \sum_{(\alpha, q)=1} \frac{\lambda(\frac{\alpha b}{(a,b)})\lambda(\frac{\alpha a}{(a,b)})}{\alpha} \Wf \left(\frac{\alpha^2
ab}{q^2(a,b)^2}\right).
\end{align}

 To evaluate the last sum above, we first recall that the Rankin-Selberg $L$-function of $L(s, f \times f)$ of $f$ is defined for $\Re(s)>1$ by (see \cite[(23.24)]{iwakow})
\begin{align*}
 L(s, f \times f)=\sum_{n \geq 1}\frac {\lambda^2(n)}{n^s}.
\end{align*}
  It is known (see \cite[p. 132]{iwakow}) that $L(s, f \times f)$ has a simple at $s=1$. In fact, we have (see \cite[(5.97)]{iwakow})
\begin{align*}
  L(s, f \times f)=\zeta(s)L(s, \operatorname{sym}^2 f),
\end{align*}
  where $L(s, \operatorname{sym}^2 f)$ is the symmetric square $L$-function of $f$ defined for $\Re(s)>1$ by (see \cite[(25.73)]{iwakow})
\begin{align*}
 L(s, \operatorname{sym}^2 f)=& \zeta(2s) \sum_{n \geq 1}\frac {\lambda(n^2)}{n^s}=\prod_{p}(1-\frac {\lambda(p^2)}{p^s}+\frac {\lambda(p^2)}{p^{2s}}-\frac {1}{p^{3s}} )^{-1}.
\end{align*}
  It follows from a result of G. Shimura \cite{Shimura} that the corresponding completed $L$-function
\begin{align*}
 \Lambda(s, \operatorname{sym}^2 f)=& \pi^{-3s/2}\Gamma (\frac {s+1}{2})\Gamma (\frac {s+\kappa-1}{2}) \Gamma (\frac {s+\kappa}{2}) L(s, \operatorname{sym}^2 f).
\end{align*}
  is entire and satisfies the functional equation
$\Lambda(s, \operatorname{sym}^2 f)=\Lambda(1-s, \operatorname{sym}^2 f)$.
 Combining this with \cite[(5.8)]{iwakow} and apply the convexity bounds (see \cite[Exercise 3, p.
  100]{iwakow}) for $L$-functions, we deduce that
\begin{align}
\label{Lsymest}
\begin{split}
  L(s, \operatorname{sym}^2 f) \ll & \left( 1+|s| \right)^{\frac {3(1-\Re(s))}{2}+\varepsilon}, \quad 0 \leq \Re(s) \leq 1.
\end{split}
\end{align}

  Note also the following convexity bound for $\zeta(s)$:
\begin{align}
\label{zetaest}
\begin{split}
  \zeta(s) \ll & \left( 1+|s| \right)^{\frac {1-\Re(s)}{2}+\varepsilon}, \quad 0 \leq \Re(s) \leq 1.
\end{split}
\end{align}

 We now evaluate the last sum in \eqref{mainterm} by setting $X=q^2(a,b)^2/(ab)$ there and applying the definition of $\Wf (x)$ given in Lemma \ref{PropDirpoly} to obtain that
\begin{align}
\label{maintermint}
& \sum_{(\alpha, q)=1} \frac{\lambda(\frac{\alpha b}{(a,b)})\lambda(\frac{\alpha a}{(a,b)})}{\alpha}  \Wf \left(\frac{\alpha^2}{X}\right)= \frac{1}{2\pi i} \int\limits_{(c)}
\frac{\Gamma\left(\frac{\kappa}{2} + s \right)^2}{(2\pi )^s \Gamma\left(\frac{\kappa}{2} \right)^2 }L(1+2s, f \times f) H(1+2s; q)G(1+2s;a,b)X^{s}
\frac{ds}{s},
\end{align}
  where $H(s;q)=\prod_{p|q}(1-\frac {\lambda^2(p)}{p^{1+2s}}+ O(\frac {1}{p^{2(1+2s)}}))$,  $G(s; a, b)=\prod_{p}G_p(s;a,b)$ with
\begin{equation*}
\begin{split}
G_{p}(s;a,b) =\left\{
 \begin{array}
  [c]{ll}
  \lambda(p^l)+ O(\frac {1}{p^{1+2s}}) & \text{if } p|  ab/(a,b)^2,\\
  1  &\text{otherwise }.
 \end{array}
 \right.
 \end{split}
\end{equation*}

  We evaluate the integral in \eqref{maintermint} by shifting the line of integration to $\Re(s)=-1/4+\varepsilon$. We encounter a double pole at $s=0$ in the
  process. Note that on the new line, we have for some constant $B_1$,
\begin{align*}
\begin{split}
 G(-\frac 14 +\varepsilon; a,b) \ll B_1^{\Omega(q)+\Omega(\frac {a}{(a,b)}+\Omega(\frac {b}{(a,b)})}d\Big(\frac {a}{(a,b)}\Big)d\Big(\frac {b}{(a,b)}\Big) \ll q^{\varepsilon},
\end{split}
\end{align*}
  where the last estimation above can be obtained using arguments that lead to \eqref{xybounds}.

  Combining the above with \eqref{Lsymest}, \eqref{zetaest} and the rapid decay of $\Gamma(s)$ when $|\Im(s)| \rightarrow \infty$, we deduce that the integration on the new line is
\begin{align*}
\begin{split}
  \ll q^{-\half+\varepsilon}.
\end{split}
\end{align*}
  Applying this in \eqref{mainterm} and taking note of the  definition of $\ell_{v+1}$ given in Section \ref{sec 2'}, we see that the contribution of the integration on the new line to the right side of \eqref{mainterm} is
\begin{align*}
\begin{split}
  \ll \phi(q)(r_k\ell_{v+1})^2 \Big( \frac{c_k r_k }{e } \Big)^{2r_k\ell_{v+1}} q^{4r_k/10^{M}-\half+\varepsilon} \ll q^{\half+4r_k/10^{M}+\varepsilon} \ll q^{1-\varepsilon}.
\end{split}
\end{align*}

   We now evaluate the corresponding residue to see that
\begin{align}
\label{alphasum}
& \sum_{(\alpha, q)=1} \frac{\lambda(\frac{\alpha b}{(a,b)})\lambda(\frac{\alpha a}{(a,b)})}{\alpha}  \Wf \left(\frac{\alpha^2}{X}\right)= C_1(q)L(1, \operatorname{sym}^2 f)G(0;a,b) \Big (\log X+2\sum_{p|ab/(a,b)^2}\frac {G'_p(1;a,b)}{G_p(1;a,b)}+C_2(q) \Big )+O( q^{-\half+\varepsilon}),
\end{align}
   where $C_1(q), C_2(q)$ are some constants depending on $q$ only, satisfying $C_i(q) \ll 1, i=1,2$.

  We apply \eqref{alphasum} to evaluate \eqref{mainterm} to see that we may ignore the contribution of the error term in \eqref{alphasum} so
  that the expression in \eqref{mainterm} is
\begin{align*}
 \ll & \phi(q) (r_k\ell_{v+1})^2 \Big( \frac{c_k r_k }{e } \Big)^{2r_k\ell_{v+1}} \\
& \times \sum_{a,b \leq q^{2r_k/10^{M}}} \frac{(a,b)}{ab} u_a u_b  G(0;a,b)\Big ( 2\log q+2 \log (a, b)-\log a -\log b+2\sum_{p|ab/(a,b)^2}\frac {G'_p(0;a,b)}{G_p(0;a,b)}+C_2(q) \Big ).
\end{align*}

  As the estimations are similar, it suffices to treat the sum
\begin{align}
\label{sumoverlog}
\begin{split}
& \phi(q)(r_k\ell_{v+1})^2 \Big( \frac{c_k r_k }{e } \Big)^{2r_k\ell_{v+1}} \sum_{a,b } \frac{(a,b)G(0;a,b)}{ab} u_a u_b \log a \\
=& \phi(q)(r_k\ell_{v+1})^2 \Big( \frac{c_k r_k }{e } \Big)^{2r_k\ell_{v+1}} \sum_{p \in \bigcup^{v+1}_{j=1} P_j}\sum_{l_1 \geq 1, l_2 \geq 0}\frac {l_1 \log p}{p^{l_1+l_2-\min (l_1, l_2)}}\sum_{\substack{ a,b \\ (ab, p)=1} } \frac{(a,b)G(0;p^{l_1}a,p^{l_2}b) u_{p^{l_1}a} u_{p^{l_2}b}}{ab} .
\end{split}
\end{align}

   We now consider the last sum above for fixed $p=p_1, l_1, l_2$. Without loss of generality, we may assume that $p_1 \in P_1$. We then define for $(n_1n'_1, p_1)=1$,
\begin{equation*}
 v_{n_1} = \frac{1}{n_1} \frac{(k-1)^{\Omega(n_1)}\widetilde{\lambda}(n_{1})}{w(n_1)}  b_1(n_1p_1^{l_1}), \quad v_{n'_1} = \frac{1}{n'_1} \frac{(k-1)^{\Omega(n'_1)}\widetilde{\lambda}(n'_{1})}{w(n'_1)}  b_1(n'_1p_1^{l_2}).
\end{equation*}
  For $2 \leq j \leq v$,
\begin{equation*}
 v_{n_j} = \frac{1}{n_j} \frac{(k-1)^{\Omega(n_j)}\widetilde{\lambda}(n_{j})}{w(n_j)}  b_j(n_j), \quad  v_{n'_j} = \frac{1}{n'_j} \frac{(k-1)^{\Omega(n'_j)}\widetilde{\lambda}(n'_{j})}{w(n'_j)}  b_j(n'_j).
\end{equation*}
  Also,
\begin{equation*}
 v_{n_{v+1}} = \frac{1}{n_{v+1}} \frac{\widetilde{\lambda}(n_{v+1})}{w(n_{v+1})}  p_{v+1}(n_{v+1}), \quad  v_{n'_{v+1}} = \frac{1}{n'_{v+1}} \frac{\widetilde{\lambda}(n'_{v+1})}{w(n'_{v+1})}  p_{v+1}(n'_{v+1}).
\end{equation*}

  Then one checks that
\begin{align}
\label{doublesum}
 \sum_{\substack{ a,b \\ (ab, p_1)=1} } \frac{(a,b)G(0;p^{l_1}a,p^{l_2}b) u_{p^{l_1}a} u_{p^{l_2}b}}{ab}=\frac {(k-1)^{l_1+l_2}\lambda^{l_1+l_2}(p_{1})G(0;p^{l_1},p^{l_2})}{l_1!l_2!}\prod^{v+1}_{j=1}\Big ( \sum_{\substack{n_j, n'_j \\ (n_1n'_1, p_1)=1}}(n_j, n'_j)G(0;n_j,n'_j)v_{n_j}v_{n'_j} \Big ).
\end{align}

   As in the proof of Proposition \ref{Prop4}, we remove the restriction of $b_1(n_1p_1^{l_1})$ on $\Omega(n_1p_1^{l_1})$
   and $b_1(n'_1p_1^{l_2})$ on $\Omega(n'_1p_1^{l_2})$ to see that the sum on the right side in \eqref{doublesum} for $j = 1$  becomes
\begin{align*}
  & \sum_{\substack{n_1, n'_1\\ (n_1n'_1, p_1)=1}}(n_1, n'_1)G(0;n_1,n'_1)\frac{\widetilde{\lambda}(n_{1})}{n_1} \frac{(k-1)^{\Omega(n_1)}}{w(n_1)} \frac{\widetilde{\lambda}(n'_{1})}{n'_1} \frac{(k-1)^{\Omega(n'_1)}}{w(n'_1)} =
  \prod_{\substack{p \in P_1 \\ p \neq p_1}}\Big ( 1+\frac{(2(k-1)+(k-1)^2)\lambda^2(p)}{p} +O(\frac 1{p^2})  \Big ) \\
\ll &  \exp (\sum_{p \in P_1}\frac{(k^2-1)\lambda^2(p)}{p}+O(\sum_{p \in P_1}\frac 1{p^2})).
\end{align*}

  Further, we notice that in this case we have $2^{\Omega(n)+l_i-\ell_1}\ge 1$ for $i=1,2$.  Thus, we apply Rankin's
  trick to see that the error introduced this way is
\begin{align*}
 \ll &  \big(2^{l_1}+2^{l_2} \big ) 2^{-\ell_1} \sum_{n_1, n'_1}\frac {(n_1, n'_1)|G(0;n_1,n'_1)|}{n_1n'_1} \frac{|k-1|^{\Omega(n_1)}|\widetilde{\lambda}(n_{1})|}{w(n_1)} \frac{|k-1|^{\Omega(n'_1)}2^{\Omega(n'_1)}|\widetilde{\lambda}(n'_{1})|}{w(n'_1)}.
\end{align*}

   We then deduce that
\begin{align*}
 &  \sum_{l_1 \geq 1, l_2 \geq 0} \frac {l_1 \log p_1}{p^{l_1+l_2-\min (l_1, l_2)}_1}\frac {(k-1)^{l_1+l_2}}{l_1!l_2!}\sum_{\substack{n_1, n'_1
 \\ (n_1n'_1, p_1)=1}}(n_1, n'_1)G(0;n_1,n'_1)v_{n_1}v_{n'_1} \\
 \ll & \sum_{l_1 \geq 1, l_2 \geq 0} \frac {l_1 \log p_1}{p^{l_1+l_2-\min (l_1, l_2)}_1} \frac {(k-1)^{l_1+l_2}}{l_1!l_2!} \Big (1+O \big ( \big (2^{l_1}+2^{l_2} \big ) 2^{-\ell_1/2} \big ) \Big )  \exp (\sum_{p \in P_1}\frac{(k^2-1)\lambda^2(p)}{p}+O(\sum_{p \in P_1}\frac 1{p^2})) \\
 \ll & \Big (1+O \big (  2^{-\ell_1/2} \big )\Big ) \Big ( \frac{\log p_1}{p_1} + O(\frac
 {\log p_1}{p^2_1}) \Big )   \exp (\sum_{p \in  P_1}\frac{(k^2-1)\lambda^2(p)}{p}+O(\sum_{p \in  P_1}\frac 1{p^2})).
\end{align*}

   Similar estimations carry over to the sums over $n_j, n'_j$ for $2 \leq j \leq v$ in \eqref{doublesum}. To treat the sum over $n_{v+1}, n'_{v+1}$,  we apply Rankin's trick again to see that the sum is
\begin{align*}
\ \ll &  (c_kr_k)^{-2r_k\ell_{v+1}} \sum_{\substack{n_{v+1}, n'_{v+1} \\ p |
n_{v+1}n'_{v+1} \implies  p\in P_{v+1} } }\frac {(n_{v+1}, n'_{v+1})|\widetilde{\lambda}(n_{v+1})\widetilde{\lambda}(n'_{v+1})G(0;n_{v+1},n'_{v+1})|}{n_{v+1}n'_{v+1}} \frac{(c_kr_k)^{\Omega(n_{v+1})}}{w(n_{v+1})} \frac{(c_kr_k)^{\Omega(n'_{v+1})}}{w(n'_{v+1})}.
\end{align*}
  By taking $N$ large enough, we deduce from this that
\begin{align*}
\begin{split}
&  (r_k\ell_{v+1})^2 \Big( \frac{c_k r_k }{e } \Big)^{2r_k\ell_{v+1}} \sum_{\substack{n_{v+1}, n'_{v+1} \\ (n_{v+1}n'_{v+1}, p_1)=1}}(n_{v+1}, n'_{v+1})v_{n_{v+1}}v_{n'_{v+1}}
\ll  e^{-\ell_{v+1}}\exp (\sum_{p \in P_{v+1}}\frac{(k^2-1)\lambda^2(p)}{p}+O(\sum_{p \in P_{v+1}}\frac 1{p^2})).
\end{split}
\end{align*}

   It follows from the above discussions that we have
\begin{align}
\label{sumpbound}
\begin{split}
&  (r_k\ell_{v+1})^2 \Big( \frac{c_k r_k }{e } \Big)^{2r_k\ell_{v+1}} \sum_{l_1 \geq 1, l_2 \geq 0}\frac {l_1 \log p_1}{p^{l_1+l_2-\min (l_1, l_2)}_1}\frac {(k-1)^{l_1+l_2}}{l_1!l_2!}\sum_{\substack{ a,b \\ (ab, p_1)=1} } \frac{u_{p^{l_1}_1a}
u_{p^{l_2}_1b}}{ab} \\
 \ll & e^{-\ell_{v+1}}\prod^v_{j=1}\big(1+O(2^{-\ell_j/2})\big)\exp (\sum_{p \in \bigcup^{v+1}_{j=1} P_j}\frac{(k^2-1)\lambda^2(p)}{p}+O(\sum_{p \in \bigcup^{v+1}_{j=1} P_j}\frac 1{p^2})) \times \Big ( \frac{\log p_1}{p_1} + O(\frac
 {\log p_1}{p^2_1}) \Big ).
\end{split}
\end{align}

  We now apply \eqref{boundsforsumoverp} and the observation $\ell_j > \ell^2_{j+1}>2\ell_{j+1}$ to see that
\begin{align}
\label{sumpbound1}
 \sum_{p \in \bigcup^R_{j=v+2} P_j}\frac{|k^2-1|\lambda^2(p)}{p} \leq \sum^{R}_{j=v+2}\sum_{p \in P_j}\frac{|k^2-1|\lambda^2(p)}{p} \leq \frac {2|k^2-1|}{N} \sum^{R}_{j=v+2} \ell_j \leq \frac {2|k^2-1|\ell_{v+2}}{N} \sum^{\infty}_{j=0} \frac 1{2^j} \leq \frac {2|k^2-1|\ell_{v+1}}{N}.
\end{align}

   It follows from \eqref{sumpbound1} that the last expression in \eqref{sumpbound} is
\begin{align*}
\begin{split}
 \ll &  e^{-\ell_{v+1}/2}\exp (\sum_{p \in \bigcup^{R}_{j=1} P_j}\frac{(k^2-1)\lambda^2(p)}{p}+O(\sum_{p \in \bigcup^{R}_{j=1} P_j}\frac 1{p^2})) \times \Big ( \frac{\log p_1}{p_1} + O(\frac
 {\log p_1}{p^2_1}) \Big ).
\end{split}
\end{align*}

 We then conclude from the above, \eqref{sumoverlog}, Lemma \ref{RS} and the observation that $\phi(q) \ll \phis(q)$ that
\begin{align*}
\begin{split}
& \sum_{\substack{ \chi \shortmod q }}|L(\tfrac{1}{2},f \otimes \chi)|^2 \Big (\prod^v_{j=1}|\mathcal{N}_j(\chi, k-1)|^2 \Big )|{\mathcal
 Q}_{v+1}(\chi, k)|^{2} \\
\ll &    \phi(q) e^{-\ell_{v+1}/2}\exp (\sum_{p \in \bigcup^{R}_{j=1} P_j}\frac{(k^2-1)\lambda^2(p)}{p}+O(\sum_{p \in \bigcup^{R}_{j=1} P_j}\frac 1{p^2})) \times \sum_{p \in \bigcup^{v+1}_{j=1}P_j} \Big ( \frac{\log p}{p} + O(\frac
 {\log p}{p^2}) \Big ) \\
\ll & \phis(q) e^{-\ell_{v+1}/2}(\log q)^{k^2}.
\end{split}
\end{align*}
  This completes the proof of the proposition.

\section{Proof of Proposition \ref{Prop6}}

   As the proofs are similar, we shall only prove here that
\begin{align*}
  \sumstar_{\substack{ \chi \shortmod q }}\prod^R_{j=1}\big ( |{\mathcal N}_j(\chi, k)|^2+ |{\mathcal Q}_j(\chi,k)|^2 \big ) \ll \phis(q)(\log q)^{ k^2 }.
\end{align*}

   We first note that
\begin{align}
\label{upperboundprodofN}
\begin{split}
 & \sumstar_{\substack{ \chi \shortmod q }}\prod^R_{j=1}\big ( |{\mathcal N}_j(\chi, k)|^2+ |{\mathcal Q}_j(\chi,k)|^2 \big )
\leq  \sum_{\substack{ \chi \shortmod q }}  \prod^R_{j=1}\big ( |{\mathcal N}_j(\chi, k)|^2+|{\mathcal Q}_j(\chi,k)|^2 \big ).
\end{split}
\end{align}

  We shall take $M$ large enough so that we may deduce from \eqref{sumoverell} that
\begin{align*}
  (2 r_k+2) \sum^R_{j=1}\frac 1{\ell_j} \leq \frac {4(r_k+1)}{\ell_R} <1 .
\end{align*}

  We then apply \eqref{5.1} and \eqref{Pexpression} in the last sum of \eqref{upperboundprodofN} and deduce from the above that
  the orthogonality relation for characters modulo $q$ implies that only the
  diagonal terms in the last sum of \eqref{upperboundprodofN} survive. Thus we obtain that
\begin{align}
\label{maintermbound}
\begin{split}
 & \sum_{\substack{ \chi \shortmod q }}  \prod^R_{j=1}\big ( |{\mathcal N}_j(\chi, k)|^2+ |{\mathcal Q}_j(\chi,k)|^2 \big )
\le  \phi(q) \prod^R_{j=1} \Big (  \sum_{n_j} \frac{k^{2\Omega(n_j)}\widetilde{\lambda}^2(n_j)}{n_j w^2(n_j)}  b_j(n_j)  + \Big( \frac{c_k} {\ell_j}\Big)^{2r_k\ell_j}((r_k\ell_j)!)^2 \sum_{ \substack{ \Omega(n_j) = r_k\ell_j \\ p|n_j \implies  p\in P_j}} \frac{\widetilde{\lambda}^2(n_j) }{n_j w^2(n_j)} \Big ).
\end{split}
\end{align}

  Arguing as before, we see that
\begin{align}
\label{sqinN}
\begin{split}
  \sum_{n_j} \frac{k^{2\Omega(n_j)}\widetilde{\lambda}^2(n_j)}{n_j w^2(n_j)}  b_j(n_j)=\Big(1+ O\big(2^{-\ell_j/2} \big ) \Big)
  \exp (\sum_{p \in P_j} \frac {k^2\lambda^2(p)}{p}+ O(\sum_{p \in P_j} \frac {1}{p^2})).
\end{split}
\end{align}
   Note also that, as $w^2(n) \geq w(n)$ and $\widetilde{\lambda}^2 (n) \geq 0$, we have
\begin{align*}
\begin{split}
 \sum_{ \substack{ \Omega(n_j) = r_k\ell_j \\ p|n_j \implies  p\in P_j}} \frac{\widetilde{\lambda}^2 (n_j)}{n_j w^2(n_j)}  \leq  \frac 1{(r_k \ell_j)!} \Big (\sum_{p \in P_j}\frac {\lambda^2(p)}{p} \Big )^{r_k\ell_j}.
\end{split}
\end{align*}

  Now, we apply the estimations given in \eqref{boundsforsumoverp} and \eqref{Stirling}  to deduce from the above that by taking $M, N$ large enough,
\begin{align}
\label{Qest}
\begin{split}
 &  \Big( \frac{c_k} {\ell_j}\Big)^{2r_k\ell_j}((r_k\ell_j)!)^2 \sum_{ \substack{ \Omega(n_j) = r_k\ell_j \\ p|n_j \implies  p\in P_j}} \frac{\widetilde{\lambda}^2(n_j) }{n_j w^2(n_j)} \ll r_k\ell_j\Big( \frac{c^2_k r_k }{e\ell_j} \Big)^{r_k\ell_j}\Big (\sum_{p \in P_j}\frac {\lambda^2(p)}{p} \Big )^{r_k\ell_j} \\
\ll &  r_k\ell_j\Big( \frac{c^2_k r_k }{e\ell_j} \Big)^{r_k\ell_j}e^{r_k\ell_j \log (2\ell_j/N)}
\ll  e^{-\ell_j} \exp (\sum_{p \in P_j} \frac {k^2\lambda^2(p)}{p}+ O(\sum_{p \in P_j} \frac {1}{p^2})).
\end{split}
\end{align}
  Applying \eqref{sqinN} and \eqref{Qest} in \eqref{maintermbound}, together with Lemma \ref{RS} and the observation that $\phi(q) \ll \phis(q)$,  we readily deduce the assertion of the proposition.

\vspace*{.5cm}

\noindent{\bf Acknowledgments.} P.G. is supported in part by NSFC grant 11871082, X. He is supported in part by NSFC grant 12101427 and X. Wu is supported in part by NSFC grant 11871187.

redpajama_set_name: RedPajamaArXiv


text:
Baylink Proposed for Miami to Miami Beach Back Under Consideration
July 21, 2014 UncategorizedHochman & Goldin PA, Miami Traffic Attorneyadmin
The cities of Miami and Miami Beach are notorious for horrible traffic jams, especially during rush hour when people are trying to get to and from work. That is why the proposed rail link between Miami and Miami Beach, recently reported in the Miami Herald, has garnered a lot of attention. Connecting these two world-class cities with a quick and modern rail system, called "Bayllink," will go a long way to improving their appeal, allowing residents and visitors alike to enjoy more of what both communities have to offer without dealing with the hassle of parking or traffic.
Although the rail link idea first emerged in the 1990s, this is the first time in nearly a decade that county commissioners have seriously looked at options for making it a reality.
On Wednesday, [the Metropolitan Planning Organization executive committee] heard from a team of consultants it has retained to study the idea. The consultants, while generally agreeing the line is feasible, sounded a symphony of cautionary notes about everything from funding to aesthetic expectations.
Bay Link, they said, would have not only too few riders to support itself (public transit systems around the country cover only about 39 percent of their costs, and Bay Link would likely do even worse) but too few to attract federal funding.
"There's intense competition [for federal transportation dollars] out there," said Robert Peskin of the consulting firm AECOM. "There are other projects that, frankly, are more likely to get the funding."
That would leave South Florida governments paying a $70 million annual tab for capital and operating expenses by themselves, he added, at a time when conventional government resources for transportation are completely tapped out.
The prospects for outside funding are so bleak, Peskin added, that officials should consider slapping a toll of $1 or $2 on every vehicle crossing the MacArthur and Julia Tuttle causeways. "I'm waiting for somebody to throw something at me," he confessed.
Even finding the right trolley designs will be complicated, as the most common systems draw their power from the their tracks and are thus impractical in light of Miami's flooding and drainage issues (the alternative – trolleys with internal power supplies – tends to be under-performing and less cost-effective).
In the end, the committee expressed enthusiasm for the consultants' proposed solution to these problems: a public-private partnership similar to the one currently working on the much anticipated Port Miami Tunnel. Such an approach could offer creative and cost-effective designs and financial solutions at relatively little cost to the county and public.
Whether the county will issue a formal bid for the project remains to be seen, but we're confident that this creative and dynamic city will find a solution. To speak to a Miami Traffic Attorney about your traffic citation, contact Hochman & Goldin PA by calling 305-665-1000.
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redpajama_set_name: RedPajamaCommonCrawl


Episode 139: Don't ask stupid questions, with Tony Parinello.
Tony Parinello created his own brand of sales training called Selling to VITO™, the Very Important Top Officer in 1995. Today, he is trusted by the majority of Fortune 1,000 and over 2.5 million sales people in more than 30 countries to help them create bigger deals in less time.
Click to tweet: Tony Parinello recommends to stop asking stupid questions on Selling Across America!
Stop asking stupid questions! Instead ask action oriented questions that will put you immediately in touch with the person who can say YES! Then following it up with a qualifying question that will get your commission check sooner not later!


text:
Encontrar y rentar espacios
Administrar propiedades y portafolio
Diseñar y concretar proyectos
Invertir en bienes raíces
Transformar con tecnología
Hoteles y hospitalidad
Tendencias y Perspectivas
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Acerca de JLL
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Seleccionar ubicación
Europa - Medio Oriente - África (EMEA)
China (Continental)
RAE Hong Kong, China
Macao RAE, China
Why some American real estate markets are more transparent than others
Even in highly transparent markets, there are differences between cities
The United States is the third most transparent real estate market in the world , topped only by the United Kingdom and Australia.
But even in highly transparent markets, there are differences between cities , according to JLL's 2018 Global Real Estate Transparency Index: City-Level Transparency report. Los Angeles, for example, is the second most transparent city in the world. Phoenix ranks 25th.
One of the big reasons these differences occur is because American state and municipal governments have autonomy to make their own rules around transaction disclosure and taxation, says Daniel Mahoney of LaSalle Investment Management.
"The patchwork of state and municipal laws make data availability differ across metropolitan areas," Mahoney says. "The transparency range in the U.S. is smaller than in many other countries, like China and Russia, thanks to high-quality data standards that are commonly applied to both tertiary and primary markets, but the divergence between U.S. markets is still often underappreciated."
A good example of local laws shaping transparency is the state of Texas, which has nondisclosure laws that prevent the sales price in real estate transactions from being public. Because only 12 states are fully nondisclosure states, overall transparency levels in the cities within them can be lower than some of their national peers.
However, in the grand scheme of things, Texas is still transparent. Two of its cities made the top 25 list of most transparent cities in the world: Dallas in the No. 20 slot and Houston in No. 24.
State laws can also increase transparency. Property taxes, for example, are more predictable in California than in many other parts of the U.S., which reduces risk, Mahoney says.
While Los Angeles is the only American city in the top three, San Francisco follows close behind in the No. 4 slot and San Diego is No. 13.
Of course, any discrepancies in the U.S. exist within a narrow spectrum, as it is still a "highly transparent" country. But as the Panama Papers in 2016 and the Paradise Papers in 2017 brought the need for greater transparency to the international stage, the differences are still worth noting, says Josh Gelormini of JLL's Americas Research.
This is especially true because a lack of transparency undermines the development of fair market structures and interferes with competition – all of which can deter investment, Gelormini says. Meanwhile, greater transparency can help to deter corruption, tax evasion and money laundering, as well as more easily shine a light on these activities when they occur despite disclosure safeguards.
"Transparency is an additional attribute that investors and corporates alike need to consider when assessing city prospects and risks," he says.
Click to find out how currency hedging is creating opportunities for U.S. investors.
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So..I have a question..I understand that the Quantum is going to use internet but does it run constantly? I only have the 200MB plan from AT&T, which is the cheapest one I could get, because -I- don't use the internet alot but..Does the phone? Am I going to have to upgrade my internet plan because the phone will use it? If so, I'm ditching this phone. I do like it but compared to my iPhone..well..It doesn't compare.
I'm on Bell here in Canada and I got a 500MB plan. I thought that it wouldn't be enough, and I was scared that the internet would be pulling data all the time, but it doesn't. I use to switch from my 3G connection to WiFi when I first got the phone, because I was afraid I was using too much data from the web. I stopped doing that though for a whole month, just to see how much data was actually pulled from the internet and I haven't even come close to my limit. I have apps that require updating over the internet and I have to say, it doesn't pull as much data as I thought.
I'm also not surfing the web all day, and the apps I have don't require too much data updating from the internet, so I'm starting to think 500MB is too much.
So to answer your question, the internet does not run constantly. If you're not using it, it won't do anything. Only if your app requires information from the internet, then you're internet will be running (for example, downloading a video, music or podcast on to your phone from your phone....and doing it everyday).
Last edited by Ivan006; 02-22-2011 at 03:12 PM.
Thanks for the info Ivan.
One of my friends at work has a 200mb plan. They are always connected to WiFi and wouldn't have a data plan if they could get away with it. Just be careful and remember some carriers dont update your current data usage right away.


In a small bowl, whisk together 4 tablespoons olive oil, vinegar, sugar, mustard, salt, and pepper to taste. Set aside.
In a large salad bowl, toss together lettuce, tomatoes, olives, and blue cheese. Set aside.
In a 12-inch non-stick frying pan, heat remaining oil over high heat.
Season steaks to taste with salt and pepper and add to pan. Cook until underside is well browned, 2-3 minutes.
Turn steaks over and cook until meat is done to your liking, about 2 more minutes for rare.
Transfer meat to a platter and let stand 2-3 minutes.
Add croutons and balsamic dressing to lettuce mixture. Toss well then divide among dinner places.
Thinly slice steaks crosswise and divide warm meat and pan juices among salads. Serve immediately.


Great, courageous and selfless leaders are those willing to confront the material anxieties of their people as the occasion arises. There is a great anxiety in Guyana as to the good life for the ordinary man as promised by the government. After three and a half years in office, the Government has spent some GYD$1.1 trillion in four consecutive budgets since 2015, and it is clear that the benefits are not trickling down to the poor and the working class. Thus, it was not surprising that the results the 2018 Local Government Elections (LGE) clearly reflected the people's disappointment with the PNC-led coalition. It was an electoral shellacking.
Today, the people are faced with a more pressing anxiety. It has been said that the health of any nation's leader is intricately intertwined with the economic health of that nation. And while we wish His Excellency a swift return to good health, the realities from the last six months have illustrated that the people are being kept in the dark. They are not being told the full truth. It is irresponsible and highly offensive for the PNC-led coalition Government to continue to attempt to play with the people's intelligence. Their public pronouncements over the last six months on how the State's business is being handled, in light of His Excellency's health, shows little respect and empathy for the people. The people are not fools and it is a vexation to their soul to observe the almost Goebel-like statements of half-truths and quarter-truths originating from the Ministry of the Presidency.
The PNC cannot abscond from its political responsibility to all the people of Guyana. What is critically needed from the PNC now is real leadership. We are therefore speaking directly to the senior leadership of the party in light of His Excellency's illness and impending absence from Guyana again. The country is in a semi-crisis state with rumors and invectives swirling from one end of the pendulum to another. It is time for Ms. Volda Lawrence and Ms. Amna Ally, the PNC's Chairman and General Secretary respectively to end the incessant speculations.
What is happening in the Ministry of the Presidency at the highest levels demands a political solution from the PNC, the principal political party in the coalition. The party stalwarts, for far too long, have ignored this issue and it has stymied national development and brought uncertainty to the country. A nation cannot advance in such an ad hoc manner.
Those at the highest levels should get out into the fields and reach out to the people so as to understand what is going on in the towns and villages. Those at the highest levels should be making the necessary but urgent decisions needed to charter the course for the good life of the people. Politics is not a game and neither is political leadership.
The PNC's leaders need to be visible, and in these times of turmoil, the nation needs leaders who are capable of making decisions in order to positively reassure the people, especially the youths, that the challenges of the times are being met. Only the leaders of the PNC are politically equipped to perform these functions since they are in Government. They can no longer be in denial and ignore the problems of the day, which remain unsolved. They must fill the political void at the apex of the Government.
The country cannot be left hanging and no one's personal interest should come before the Country's. That is why we are calling on both Ms Lawrence and Ms Ally to step up and solve the leadership problem. To do otherwise would result in a lack of public confidence in the Government.
The PNC senior leaders must apprise the nation of their plans for the near future around this important issue of leadership succession.
In spite of His Excellency being ill, they must take the bull by the horns and put forth a clear picture of what they intend to do in these tumultuous times. The nation is in pain and our youths are suffering and are crying out for help. This is primarily because the PNC as a party lacks effective leadership to deal with the issues at hand. This must change. It is time for the PNC senior leaders to step up or get out of the kitchen.
God bless His Excellency, President David Granger and we wish him a swift return to good health and God bless our beloved Guyana.