Universality and distribution of zeros and poles of some zeta functions

Abstract

This paper studies zeta functions of the form P∞ n=1 χ(n)n −s , with χ a completely multiplicative function taking only unimodular values. We denote by σ(χ) the infimum of those α such that the Dirichlet series P∞ n=1 χ(n)n −s can be continued meromorphically to the halfplane Res > α, and denote by ζχ(s) the corresponding meromorphic function in Res > σ(χ). We construct ζχ(s) that have σ(χ) ≤ 1/2 and are universal for zero-free analytic functions on the halfcritical strip 1/2 < Res < 1, with zeros and poles at any discrete multisets lying in a strip to the right of Res = 1/2 and satisfying a density condition that is somewhat stricter than the density hypothesis for the zeros of the Riemann zeta function. On a conceivable version of Cramér’s conjecture for gaps between primes, the density condition can be relaxed, and zeros and poles can also be placed at β+ iγ with β ≤ 1− λlog log|γ|/log|γ| when λ > 1. Finally, we show that there exists ζχ(s) with σ(χ) ≤ 1/2 and zeros at any discrete multiset in the strip 1/2 < Res ≤ 39/40 with no accumulation point in Res > 1/2; on the Riemann hypothesis, this strip may be replaced by the half-critical strip 1/2 < Res < 1.