Applications of a theorem of H. Cramér to the Selberg class

Abstract:

We prove two results on the nature of the Dirichlet coefficients $a(n)$ of the $L$-functions in the extended Selberg class $\mathcal {S}^\sharp$. The first result asserts that if $a(n)=\phi (\log n)$ for some entire function $\phi (z)$ of order 1 and finite type, then $\phi (z)$ is constant. The second result states, roughly, that if $a(n)\phi (\log n)$ are still the coefficients of some $L$-function from $\mathcal {S}^\sharp$, then $\phi (z)=ce^{i\beta z}$ with $c\in \mathbb {C}$ and $\beta \in \mathbb {R}$. The proofs are based on an old result by Cramér and on the characterization of the functions of degree 1 of $\mathcal {S}^\sharp$.