# Tagged Questions

0answers
88 views

### Identity involving $\zeta(3)$

This is a follow-up of this question and my partial answer to it. I've found that the proof of the 2nd identity reduces to showing that ...
2answers
239 views

### Clausen and Riemann zeta function

This is an exercise from the American Monthly Problems from last year. I would like prove two formulas: (1) \$\int_0^{2\pi}\int_0^{2\pi}\log(3+2\cos(x)+2\cos(y)+2\cos(x-y)) dxdy=8\pi ...
0answers
35 views

### The minimum of a function

Could anyone possibly give me any help with finding the minimum of this function? I believe the result to be $2\pi |n|$ from page 619 of this paper by W. G. C. Boyd. \begin{equation} ...
1answer
77 views

### How does one calculate the amount of time required for computation?

For example, to compute the zeroes of the Riemann zeta function using the Euler-Maclaurin summation method one has to do O(T) work. The Euler-Maclaurin summation method for zeta is given by \$ ...
2answers
191 views

### How to find integral of $\int_0^\infty \frac{\ln ^2z} {1+z^2}\mathrm{d}z$?

How do I find the value of $$\int_{0}^{\infty} \frac{(\ln z)^2}{1+z^2}\mathrm{d}z$$ without using contour integration, - using the usual special functions, e.g., zeta/gamma/beta/etc. Thank you,
0answers
267 views

### An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found  P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n} ...
1answer
92 views

### Riemann's Zeta function [duplicate]

Possible Duplicate: Riemann Zeta Function and Analytic Continuation Calculating the Zeroes of the Riemann-Zeta function It is stated that Riemann's Zeta function has zeros at negative ...
1answer
162 views

### How does $\zeta(1 - s)$ become $(-1/s + \cdots)$?

On the Wiki page on $1 + 1 + \cdots$, how does $\zeta(1 - s)$ become $(-\frac{1}{s} + \cdots)$? A detailed explanation would be appreciated.
2answers
179 views

### Detailed proof of $\zeta(s)-1/(s-1)$ extends holomorphically to $\Re(s)>0$

I'm trying to understand the proof of PNT by Don Zagier. But his proof is too simplified so I can't understand it. I got stumped at step II: $\zeta(s)-1/(s-1)$ extends holomorphically to ...
3answers
395 views

### Derivatives of the Riemann zeta function at $s=0$

It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n!}$, among other results on $\zeta^{(n)}(0)$ . the sequence : \delta_{n}=\left | ...
0answers
75 views

### fastest way to evaluate $\arg\zeta\left(\frac{1}{2}+i\text{t}\right)$ [duplicate]

Possible Duplicate: evaluation of $\operatorname{Arg}\zeta (1/2+is)$ ?? If we consider \arg\zeta\left(\frac{1}{2} + i\text{t}\right) = \text{Im ...
1answer
182 views

### Improper integral about exp appeared in Titchmarsh's book on the zeta function

May I ask how to do the following integration? $$\int_0^\infty \frac{e^{-(\pi n^{2}/x) -(\pi t^2 x)}}{\sqrt{x}} dx$$ where $t>0$, $n$ a positive integer. This came up on page 32 (image) of ...
0answers
64 views

### Is this formula for $\sum_{n} (n^{2}+z^{2})^{-s}$ correct?

I would like to know if this formula is true: $$\sum_{n=1}^{\infty}\frac{1}{(z^{2}+n^{2})^s}=\frac{1}{\Gamma(s)} \sum_{n=0}^{\infty}\Gamma(s+n)\zeta(2s+2n)\frac{ (-z^2)^n}{n!}.$$ I have used the ...
2answers
92 views

### Riemann zeta sums and harmonic numbers

Given the nth harmonic number of order s, $$H_n(s) =\sum_{m=1}^n \frac{1}{m^s}$$ It can be empirically observed that, for $s > 2$, then, \sum_{n=1}^\infty\Big[\zeta(s)-H_n(s)\Big] = ...
1answer
208 views

### On the zeta sum $\sum_{n=1}^\infty[\zeta(5n)-1]$ and others

For p = 2, we have, $\begin{align}&\sum_{n=1}^\infty[\zeta(pn)-1] = \frac{3}{4}\end{align}$ It seems there is a general form for odd p. For example, for p = 5, define $z_5 = e^{\pi i/5}$. Then, ...
1answer
214 views

### What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?

We have the following evaluations: \begin{aligned} &\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\ &\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = ...
1answer
146 views

### Limit of the function $\zeta(x)/\zeta(x+1)$ as $x \to \infty$

I am looking for a simple proof that $\zeta(\alpha)/\zeta(\alpha+1) \to 1$ as $\alpha \to \infty$ (where $\zeta(\alpha)$ denotes the Riemann zeta function, \$\zeta(\alpha) = \sum \limits_{n\geq 1} ...
2answers
990 views

### Proving an “amazing” claim regarding $\zeta( 3)$ and Apéry's proof

I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$ $$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$ and thus (probably) ...
2answers
226 views

### Evaluating $\zeta(0)$ using the functional equation of Riemann-Zeta function.

$$\zeta(it)=2it\pi it−1\sin(i\pi t/2)\Gamma(1−it)\zeta(1−it).$$ Everything on the RHS is never zero, Does that means LHS has no zeros, since $\sin(s)$ has a simple zero at $s=0$ while $\zeta(1−s)$ ...
0answers
147 views

### How is the Riemann-Siegel formula applied?

What is the application of the Riemann-Siegel formula: $$\zeta(s) = \sum_{n=1}^N\frac{1}{n^s} + \gamma(1-s)\sum_{n=1}^M\frac{1}{n^{1-s}} + R(s) ,$$ where \$ \displaystyle\gamma(s) = ...
1answer
149 views

### Zeta function identity

How does one prove the zeta function identity $$\sum_{s=2}^{\infty}\left(1-\sum_{n=1}^{\infty}\frac{1}{n^s}\right)=-1 \;?$$
1answer
195 views

### Verifying identities for Riemann zeta function

I ran across these two problems, while reading a text on number theory. The problem states: "Verify the following identities". What does "verify" mean in this context, and what strategies can I employ ...
3answers
279 views

### Limit of Zeta function

I'm looking for a reference for (or an elementary proof of) $$\lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$ Thanks for your help.
1answer
452 views

### Upper bound on differences of consecutive zeta zeros

The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann's zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many ...
2answers
162 views

### Approximate Riemann zeta function

Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$. In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function. My question is: Is there a Functional equation for ...
0answers
174 views

### New generalization of Riemann Zeta?

I am interested in the following generalization of the Riemann Zeta function: $$\zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s}$$ This is most closely related ...
2answers
599 views

### Logarithmic derivative of Riemann Zeta function

Given the logarithmic derivative of the zeta function $\dfrac{\zeta^\prime (s)}{\zeta(s)}$ how does it behave near $s=1$? I mean if for $s=1$ the Laurent series for the logarithmic derivative becomes ...
1answer
315 views

### elliptic generalizations of Euler's trick

So Euler employed the following identity $$\sin(z) = z \prod_{n=1}^{\infty} \left[1-\left(\frac{z}{n\pi}\right)^{2}\right]$$ to evaluate $\zeta(2n)$, for $n\in\mathbb{N}$ I'm curious if there's been ...
3answers
1k views

### Analytic continuation- Easy explanation?

Today, as I was flipping through my copy of Higher Algebra by Barnard and Child, I came across a theorem which said, The series $$1+\frac{1}{2^p} +\frac{1}{3^p}+...$$ diverges for $p\leq 1$ and ...
2answers
356 views

### Integrating $\frac{x^k }{1+\cosh(x)}$

In the course of solving a certain problem, I've had to evaluate integrals of the form: $$\int_0^\infty \frac{x^k}{1+\cosh(x)} \mathrm{d}x$$ for several values of k. I've noticed that that, for k a ...
3answers
305 views

### Erroneous numerical approximations of $\zeta\left(\frac{1}{2}\right)$?

By definition of the Riemann Zeta Function, $$\zeta\left(\frac{1}{2}\right) = \sum_{n=1}^\infty \frac{1}{\sqrt{n}}.$$ Since $\forall n \geq 1 : \frac{1}{\sqrt{n}} \geq \frac{1}{n}$, we have that for ...
2answers
347 views

### Is it problem of Mathematica or my own?

The following is a plot comparing Exp[Derivative[1,0][Zeta][0,x]+1/2Log[2 Pi]] and Gamma[x]: In theory the blue and the red ...
4answers
1k views

### Proving a known zero of the Riemann Zeta has real part exactly 1/2

Much effort has been expended on a famous unsolved problem about the Riemann Zeta function $\zeta(s)$. Not surprisingly, it's called the Riemann hypothesis, which asserts:  \zeta(s) = 0 ...
4answers
473 views

### Are there addition formulas for the Riemann Zeta function?

In particular for two real numbers $a$ and $b$, I'd like to know if there are formulas for $\zeta (a+b)$ and $\zeta (a-b)$ as a function of $\zeta (a)$ and $\zeta (b)$. The closest I could find ...
2answers
1k views

### How to evaluate Riemann Zeta function

How do I evaluate this function for given $s$? $$\zeta(s) = \sum_{n=1}^\infty \frac1{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$$
2answers
505 views

### Question Relating Gamma Function to Riemann Zeta function evaluated at integers

I was just reading a paper of Ramanujan entitled " On question 330 of Professor Sanjana" when i got stuck up with a Proposition which i am unable to answer. The proposition is if \$ \displaystyle ...
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