Zeros of polynomials of derivatives of zeta functions

Abstract

Let $P_s\in {\mathcal{D}}_s[X_0,X_1,\dots ,X_l]$ 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 $P_s(L(s),L^{(1)}(s),\dots ,L^{(l)}(s))$ 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 $P_s\in {\mathcal{D}}_s[X_0,X_1,\dots ,X_l]$ is a polynomial such that at least one of the degree of $X_1,\dots ,X_l$ is greater than zero. As a corollary, we prove that the function $(d^k/d{s}^k)P_s(L(s))$ 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 $P_s(L(s),L^{(1)}(s),\dots ,L^{(l)}(s))$ and $(d^k/d{s}^k)P_s(L(s))$ are studied as well.

10 pages