Selberg's orthonormality conjecture states that if $F$ and $G$ are primitive functions of the Selberg class, then one has:
$\displaystyle{\sum_{p\leq x}\dfrac{a_{p}(F)\overline{a_{p}(G)}}{p}}=\delta_{FG}\log\log x+O(1)$
Where $\delta_{FG}=1$ if and only $F=G$ and $\delta_{FG}=0$ otherwise.
On the other hand, the orthogonality theorems for characters of a finite group $G$ states that:
$\displaystyle{\sum_{g\in G}\chi_{1}(g)\overline{\chi_{1}(g)}}=\mid G\mid$
while
$\displaystyle{\sum_{g\in G}\chi_{1}\overline{\chi_{2}(g)}}=0$
where $\chi_{1}$ and $\chi_{2}$ are the characters of two inequivalent irreducible linear representations of $G$.
So my question is: can $\dfrac{a_{p}}{\sqrt{p}}$ be interpreted as a character of a linear representation of a finite group of order $\log\log x+O(1)$?
Thanks in advance.
I don't see why you could possibly expect to interpret $a_p/\sqrt{p}$ -- a number -- as a character of some finite group. Not to mention that asking that a finite group be of order $\log\log x+O(1)$ is completely meaningless. Being $O(1)$ is a property of a function, not a number (the order of the group).
Is there any reason for this other than that both results/conjectures contain the word "orthogonality"/"orthonormality" in their names? That should hardly be surprising; orthonormality appears everywhere. One result says that irreducible characters form an orthonormal basis of the space of class functions. Selberg's conjecture says (something like) that primitive L-functions form an orthogonal basis of the Selberg class.
Would you expect, say, for there to be a link between the orthogonality of irreducible characters and the fact that there is an orthogonal basis of some given Hilbert space?