# 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?

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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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 ;) – InvisiblePanda Jan 17 '13 at 18:10
Thank you for your comment. I changed the title. – Sylvain Julien Jan 17 '13 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 '13 at 19:22
I cannot see what is the purpose of including your conjectures and what not. This is not the correct site to post conjectures on how to prove GRH and ideas for their proofs... – Mariano Suárez-Alvarez Feb 1 '15 at 10:29

$SG_{A}=PG_{A}/[Id]$, 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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