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 $\delta >0$, 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 ${\sum }_{p\le x}|a(p){|}^2\sim \kappa \pi (x)$ 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.

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

References in corpus (3)

• Zeros of L-functions outside the critical strip
• Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line
• Numerical computations on the zeros of the Euler double zeta-function I