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

TL;DR

This paper proves that derivatives of logarithms of certain $L$-functions from the Selberg class take on all values infinitely often in the half-plane of absolute convergence, revealing rich value distribution properties.

Contribution

It establishes the infinite value distribution of derivatives of $ ext{log} \, ext{L}$-functions in the half-plane of absolute convergence for the first time.

Findings

$( ext{log} \, ext{L}(s))^{(m)}$ attains all values infinitely often in the strip $1< ext{Re}(s)<1+ ext{delta}$

$ ext{L}(s)$ takes all non-zero values infinitely often in the same strip

The first derivative of $ ext{L}(s)$ vanishes infinitely often

Abstract

In the present paper, we show that, for every $\delta>0$, 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 $1<\Re(s) <1+\delta$, provided $\sum_{p\leq x} |a (p)|^2 \sim \kappa\pi(x)$ for some $\kappa>0$. In particular, ${\mathcal{L}} (s)$ takes any non-zero value infinitely often in the strip $1<\Re(s)<1+\delta$, and the first derivative of ${\mathcal{L}} (s)$ vanishes infinitely often.