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

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 $\mathcal{S}^\sharp$.