Zeros of polynomials of derivatives of zeta functions

Abstract

Let P s ∈ D s [ X 0 , X 1 , … , X l ] 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 ℜ ( s ) > 1 / 2 \Re (s) > 1/2 . In the present paper, we show that the function P s ( L ( s ) , L ( 1 ) ( s ) , … , L ( l ) ( s ) ) P_s(L(s), L^{(1)}(s),\ldots , L^{(l)}(s)) has infinitely many zeros in the vertical strip D := { s ∈ C : 1 / 2 > ℜ ( s ) > 1 } D:= \{ s \in {\mathbb {C}} : 1/2 > \Re (s) >1\} if L ( s ) L(s) is hybridly universal and P s ∈ D s [ X 0 , X 1 , … , X l ] P_s \in \mathcal {D}_s[X_0,X_1, \ldots ,X_l] is a polynomial such that at least one of the degrees of X 1 , … , X l X_1,\ldots ,X_l is greater than zero. As a corollary, we prove that the function ( d k / d s k ) P s ( L ( s ) ) (d^k / ds^k) P_s(L(s)) with k ∈ N k \in {\mathbb {N}} has infinitely many zeros in the strip D D when L ( s ) L(s) is hybridly universal and P s ∈ D s [ X ] 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 ) , … , L ( l ) ( s ) ) P_s(L(s), L^{(1)}(s),\ldots , L^{(l)}(s)) and ( d k / d s k ) P s ( L ( s ) ) (d^k / ds^k) P_s(L(s)) are studied as well.