Question

Hello, I would like to know which results have been obtained concerning Selberg's orthonormality conjecture. For example, has it been proven that for every pair of distinct primitive functions of the Selberg class $(F,G)$, $\displaystyle{\sum_{p\leq x}\frac{a_p(F)\overline{a_p(G)}}{p}=o(\log\log x)}$? Thank you in advance.

Answer 1

Orthogonality has been proven for certain pairs of automorphic L-functions. The proof procedes like a proof of the Prime Number Theorem, replacing the logarithmic derivative of $\zeta(s)$ with that of $L(s,\pi\otimes\tilde\pi')$, where $\pi$ and $\pi'$ are automorphic representations.
The only proofs in print are for pairs of representations on $GL_m$ (not necessarily the same $m$ for both representations) over ${\mathbb Q}$. It looks like it applies to any pair where the Rankin-Selberg L-function satisfies:
1) Absolutely convergent Euler product for Re(s)$>1$.
2) The expected meromorphic continuation and functional equation, with at most one simple pole.
3) Some estimate towards the Ramanujan conjecture.
4) Nonvanishing for Re(s)$\ge 1$.
5) Order 1 (in the sense that $\log|f(z)|\ll |z|^{1+\epsilon}$), away from the possible pole.
It would be very exciting if nonvanishing (or order 1) could be proven from the Selberg Class axioms. Without them (i.e. proving orthogonality for the entire Selberg Class), the above proof fails completely, so something entirely different would be necessary.

Ye has the above proof on his website, assuming one of the representations is self-dual. The above proof without the assumption can be found here.