Hello,


I read quite quickly an article in which it is shown that the unicity of factorization in the Selberg class S is equivalent to some kind of linear independence of distinct primitive functions of S. So my question is: is this unicity of factorization in fact equivalent to Selberg's orthonormality conjecture?
Thank you in advance.



Is the unique factorization in S equivalent to SOC?
Either the Selberg class is not family of automorphic $L$-functions on $GL(n)$, in which case all bets are off, or it is, and then both facts are very probably true, and therefore logically equivalent.


On the other hand, if we take as "model" for $L$-functions the representations of a group, even for a finite group, we see that unique factorization is similar to uniqueness of irreducible factors in a composition series, while orthonormality is analogue to Schur's Lemma over an algebraically closed field. Both are true, but they don't feel as "intuitively" equivalent to me -- e.g., because the former, or even Maschke's stronger semisimplicity theorem, holds over any field of characteristic zero (say), whereas Schur's Lemma is quite different over a non-algebraically closed field.


