Value distribution for the derivatives of the logarithm of $L$-functions from the Selberg class in the half-plane of absolute convergence

Abstract

In the present paper, we show that, for every $δ>0$, the function $(\log {\mathcal{L}}(s))^{(m)}$, where $m\in {\mathbb{N}} \cup \{ 0\}$ and ${\mathcal{L}} (s) := \sum_{n=1}^\infty a(n) n^{-s}$ is an element of the Selberg class ${\mathcal{S}}$, takes any value infinitely often in any strip $1<\Re(s) <1+δ$, provided $\sum_{p\leq x} |a (p)|^2 \sim κπ(x)$ for some $κ>0$. In particular, ${\mathcal{L}} (s)$ takes any non-zero value infinitely often in the strip $1<\Re(s)<1+δ$, and the first derivative of ${\mathcal{L}} (s)$ vanishes infinitely often.

12 pages. We rewrote the Proof of Corollary 1.6 and deleted Lemma 2.7 since we do not need Lemma 2.7 in the Proof of Corollary 1.6