I'm currently working on a conditional proof of the Grand Riemann Hypothesis, which is based on the assumption that every field automorphism of $\mathbb{C}$ that commutes with an element of the Selberg class is continuous. It seems that Selberg's orthogonality conjecture (SOC for short) implies such a continuity condition, judging by the following, which is a cross-post from MSE:


Indeed let $F$ be an element of the Selberg class and $\sigma$ a field automorphism of $\mathbb{C}$ that commutes with $F$. Let $\Phi\_{\sigma}:F\mapsto \sigma\circ F\circ\sigma^{-1}$. Assuming $\Phi\_{\sigma}$ maps a primitive element of the Selberg class to a primitive element of the Selberg class and Selberg's orthogonality conjecture, one gets that $\langle F,G\rangle=\langle \Phi\_{\sigma}(F),\Phi\_{\sigma}(G)\rangle$ where the inner product $\langle F,G\rangle$ is defined on the space $\mathcal{J}$ (which is defined as the complex linear space generated by the Selberg class) as the limit as $x$ tends to infinity of $\dfrac{1}{\log \log x}\sum\_{p\le x}\dfrac{a\_{p}(F)\overline{a\_{p}(G)}}{p}$ with $a\_{p}(F)$ such that for $\Re(s)>1$, $F(s)=\sum\_{n>0}{\dfrac{a\_{n}(F)}{n^s}}$. This makes $\Phi\_{\sigma}$ continuous (more precisely, an automorphism of $\mathcal{J}$).


Now, writing $\langle F+\epsilon F,F+\epsilon F\rangle=\langle \Phi\_{\sigma}(F+\epsilon F),\Phi\_{\sigma}(F+\epsilon F)\rangle$, one gets $\langle \sigma\circ(F+\epsilon F)\circ\sigma^{-1},\sigma\circ(F+\epsilon F)\circ\sigma^{-1}\rangle=\langle F+\epsilon F, F+\epsilon F\rangle$, hence $\langle \sigma\circ F\circ\sigma^{-1}+\sigma\circ\epsilon F\circ\sigma^{-1},\sigma\circ F\circ\sigma^{-1}+\sigma\circ\epsilon F\circ\sigma^{-1}\rangle=\langle F+\epsilon F,F+\epsilon F\rangle$, thus $\langle\sigma(\epsilon)F,\sigma(\epsilon)F\rangle=\langle \epsilon F,\epsilon F\rangle$. Taking the limit of both sides of the equality as $\epsilon$ tends to $0$, one gets that $\lim\_{\epsilon\to 0}\sigma(\epsilon)=0$, hence making $\sigma$ continuous. 


My first question is: is this correct?  

My second question is: is SOC already known to imply the Grand Riemann Hypothesis? If so, could I get some reference?  

Thanks in advance.



Is SOC known to imply the Grand Riemann Hypothesis?
It is not known that SOC implies GRH. In fact it is not even known that every automorphic $L$-function (which is the subject of GRH) belongs to the Selberg class (because the Ramanujan-Selberg conjecture is an open problem).


The rest of your post seems off-topic: MO is not for verifying mathematical arguments. For that purpose there are the journals, arXiv, blogs, email, etc.


