Zeros of polynomials of derivatives of zeta functions

TL;DR

This paper investigates the zeros of polynomial combinations of derivatives of zeta functions, showing they have infinitely many zeros in a specific strip when the functions are hybridly universal, and provides bounds on their number.

Contribution

It establishes the existence of infinitely many zeros for polynomial derivatives of zeta functions under hybrid universality and analyzes zero bounds.

Findings

Infinitely many zeros in the strip 1/2 < Re(s) < 1 for certain polynomial combinations.

Zero bounds for these polynomial functions and their derivatives.

Extension to derivatives of polynomials of L-functions.

Abstract

Let $P_s \in \mathcal{D}_s[X_0,X_1, \ldots,X_l]$ be a polynomial whose coefficients are the ring of all general Dirichlet series which converge absolutely in the half-plane $\Re (s) > 1/2$. In the present paper, we show that the function $P_s(L(s), L^{(1)}(s),\ldots, L^{(l)}(s))$ has infinitely many zeros in the vertical strip $D:= \{ s \in {\mathbb{C}} : 1/2 < \Re (s) <1\}$ if $L(s)$ is hybridly universal and $P_s \in \mathcal{D}_s[X_0,X_1, \ldots,X_l]$ is a polynomial such that at least one of the degree of $X_1,\ldots,X_l$ is greater than zero. As a corollary, we prove that the function $(d^k / ds^k) P_s(L(s))$ with $k \in {\mathbb{N}}$ has infinitely many zeros in the strip $D$ when $L(s)$ is hybridly universal and $P_s \in \mathcal{D}_s[X]$ is a polynomial with degree greater than zero. The upper bounds for the numbers of zeros of $P_s(L(s), L^{(1)}(s),\ldots, L^{(l)}(s))$ and $(d^k / ds^k) P_s(L(s))$ are studied as well.