Zeros of polynomials of derivatives of zeta 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.