Hello,

assuming Selberg's orthonormality conjecture, let's consider bijective maps f from Selberg's class to itself such as:

1. f maps a primitive function of S to a primitive function of S,

2. f maps a function of degree d to a function of degree d,

3. for every F in S, n_f(F)=n_F, where n_F is the integer involved in Selberg's conjecture A,

4. if F=F_{1}^{e_{1}}.F_{2}^{e_{2}}....F_{k}^{{e_{k}}, then f(F)=f(F_{1})^{e_{1}}.f(F_{2})^{e_{2}}....f(F_{k})^{{e_{k}} (so that f is "strongly multiplicative"),

5. if f verifies the above conditions, then so does the inverse of f.

The set of all such maps make a group for the composition law. I would like to know whether this group is isomorphic to the absolute Galois group of the field of the rational numbers or not.

Thanks in advance.

Hello,

assuming Selberg's orthonormality conjecture, let's consider bijective maps $f$ from Selberg's class $\mathcal{S}$ to itself such as:

1. $f$ maps a primitive function of $\mathcal{S}$ to a primitive function of $\mathcal{S}$,

2. $f$ maps a function of degree $d$ to a function of degree $d$,

3. for every $F$ in $\mathcal{S}$, $n_{f(F)}=n_{F}$, where $n_{F}$ is the integer involved in Selberg's conjecture $A$,

4. if $F=F_{1}^{e_{1}}.F_{2}^{e_{2}}....F_{k}^{e_{k}}$, then $f(F)=f(F_{1})^{e_{1}}.f(F_{2})^{e_{2}}....f(F_{k})^{e_{k}}$ (so that $f$ is "strongly multiplicative"),

5. if $f$ verifies the above conditions, then so does the inverse of $f$.

The set of all such maps makes a group for the composition law. I would like to know whether this group is isomorphic to the absolute Galois group of the field of the rational numbers or not.

Thanks in advance.

Hello,

assuming Selberg's orthonormality conjecture, let's consider maps f from Selberg's class to itself such as:

1. f maps a primitive function of S to a primitive function of S,

2. f maps a function of degree d to a function of degree d,

3. for every F in S, n_f(F)=n_F, where n_F is the integer involved in Selberg's conjecture A,

4. if F=F_{1}^{e_{1}}.F_{2}^{e_{2}}....F_{k}^{{e_{k}}, then f(F)=f(F_{1})^{e_{1}}.f(F_{2})^{e_{2}}....f(F_{k})^{{e_{k}} (so that f is "strongly multiplicative"),

5. if f verifies the above conditions, then so does the inverse of f.

The set of all such maps make a group for the composition law. I would like to know whether this group is isomorphic to the absolute Galois group of the field of the rational numbers or not.

Thanks in advance.

Hello,

assuming Selberg's orthonormality conjecture, let's consider bijective maps f from Selberg's class to itself such as:

1. f maps a primitive function of S to a primitive function of S,

2. f maps a function of degree d to a function of degree d,

3. for every F in S, n_f(F)=n_F, where n_F is the integer involved in Selberg's conjecture A,

4. if F=F_{1}^{e_{1}}.F_{2}^{e_{2}}....F_{k}^{{e_{k}}, then f(F)=f(F_{1})^{e_{1}}.f(F_{2})^{e_{2}}....f(F_{k})^{{e_{k}} (so that f is "strongly multiplicative"),

5. if f verifies the above conditions, then so does the inverse of f.

The set of all such maps make a group for the composition law. I would like to know whether this group is isomorphic to the absolute Galois group of the field of the rational numbers or not.

Thanks in advance.