# Definition of a peculiar quotient group of isometries

I just wrote a text file to sum up my ideas about the Riemann Hypothesis. This text is a draft, and I don't expect people here to say if this approach is interesting or not (but if by chance you think so, please let me know). My question is about the definition of a peculiar quotient group, which might not be correct in definition 6.
Can someone tell me if it's ok or not?
Thank you in advance.

Here comes the text:

Main ideas towards a proof of GRH

Definition 1
Let $A$ be be a subclass of the Selberg class containing $1$, closed under products, and such that every element of $A$ can be factored in a unique fashion in a product of primitive elements of $S$, these primitive elements belonging to $A$. Such a subclass of $S$ will be called a Galois class of L-functions.

Definition 2
Let $A$ be a Galois class of L-functions. $T$ is an automorphim of $A$ iff the following properties simultaneously hold true:

1) $T$ is a bijective map from $A$ to itself
2) $T$ maps a primitive element of $A$ to a primitive element of $A$
3) for all $F$ in $A$, the degree of $F$ and the degree of $T(F)$ are the same
4) for all $F$, $G$ in $A$, $T(F.G)=T(F).T(G)$

Definition 3
$\tilde{\phi}:F\mapsto \phi\circ F\circ \phi^{-1}$ is said to be an isometric automorphism of $A$ if $\tilde{\phi}$ is an automorphism of $A$ and $\phi$ is an isometry of the complex plane preserving any rectangle centered in $1/2$.

Proposition 1
$\tilde{\phi}(F)=F$ implies $\phi(0)=0$.

Definition 4
A group of equivalence classes of isometries is said to be nice iff each equivalence class contains exactly one representant $\phi$ such that $\phi(0)=0$.

Let $F\in A$ and let $G_F$ be the group of all equivalence classes of isometries preserving the set of non trivial zeros of $F$, where the equivalence relation $R_F$ is defined by : $f R_F g$ iff for every non-trivial zero $s$ of $F$, $f(s)=g(s)$.

Let $G'_F$ be the group of isometric automorphisms of $A$ preserving $F$.

Proposition 2
$G_F$ and $G'_F$ are isomorphic iff $G_F$ is nice.

Definition 5

Let $A$ and $B$ two Galois classes of L-functions such that $A$ is a sub Galois class of L-functions of $B$. Let's define the L-type (Isometric) Structure group of $B$ over $A$ as the group of (isometric) automorphisms of $B$ preserving $A$ pointwise. The L-type structure group of $B$ over $A$ will be denoted as LStr(B/A) and the L-type Isometric Structure Group of $B$ over $A$ LIStr(B/A).

Definition 6

Let $GZ(A)$ be the set $\bigcup_{F\in A}\{Z(F):=\{s,F(s)=0, 0<\Re(s)<1\}\}$. The group of all isometries preserving $GZ(A)$ will be denoted as $PG_A$. Now let's consider the relation $R'_A$ defined by $f R'_A g,$, where $f$, $g$ are elements of $PG_A$, iff for all $F$, $H$ in $A$, $f R_F g$ iff $f R_H g$. The quotient group $PG_A/R'_A$ will be denoted as $SG_A$. Let's now consider the group of elements of $SG_B$ preserving $GZ(A)$ pointwise. This last group will be called "Z-type Isometric Structure Goup of $B$ over $A$" and denoted as ZIStr(B/A).

Conjecture 1
Let $A$ be a Galois class of L-functions, $F$ an element of $A$. $G_F$ is nice iff for all $H$ in $A$, $G_H$ is nice.

Conjecture 2 (Main Conjecture)
Let $A$ and $B$ be two Galois classes of L-functions such that $A$ is a sub-Galois class of L-functions of $B$. Then LIStr(B/A) and ZIStr(B/A) are isomorphic.

Idea of proof: prove that LIStr(B/A) and ZIStr(B/A) are both isomorphic to $Gal(F_{B}/F_{A})$, where $F_B$ is the field generated by the union of all images of $\mathbb{R}-\{1\}$ by the elements of $B$, and $F_{A}$ is defined in a similar way. If $B$ is the maximal Galois class of L-functions and $A=\{\zeta^{n}, n\geq 0\}$, one can expect to have a proof of the Riemann Hypothesis.

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1
You should maybe think about changing the title to something more specific. I for one came here thinking this question is about the definition of 'quotient groups' and nothing too advanced ;) – Rand al'Thor Jan 17 at 18:10
Thank you for your comment. I changed the title. – Sylvain Julien Jan 17 at 18:33
Not exactly an answer to the initial question, but one can show that $F_A=\mathbb{R}$ if and only if all the elements of $A$ are self-dual, otherwise $F_{A}=\mathbb{C}$. Hope this can be of interest... – Sylvain Julien Jan 21 at 19:22

## 1 Answer

$SG_{A}=PG_{A}/[Id]$, where $[Id]$ is the equivalence class of the neutral element of $PG_{A}$ for the relation $R'_{A}$. So that $SG_{A}$ is well defined.

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