Hello,

let $F$ and $G$ be two primitive functions of the Selberg class, and let $\mathbb{A}$ be the set of values taken by the function which maps a prime number $p$ to $a_{p}(F)\overline{a_{p}(G)}$. $\mathbb{A}$ is finite or countable, so that there exists $I\subset\mathbb{N}^*$ such that $\displaystyle{\mathbb{A}=\bigcup_{i\in I}\{A_{i}\}}$, with $A_i\neq A_j$ whenever $i\neq j$. Now, let $\delta$ be the function that maps $i\in I$ to $\displaystyle{\lim_{x\to\infty}\frac{#\{p\leq x, a_{p}(F)\overline{a_{p}(G)}=A_i\}}{\pi(x)}}$.

Does Selberg's orthonormality conjecture imply $\displaystyle{\sum_{i}\delta(i)A_i=\delta_{F,G}}$, where $\delta_{F,G}=1$ if $F=G$ and $\delta_{F,G}=0$ otherwise? If so, is such an implication in fact an equivalence? Thank you in advance.