Universality and distribution of zeros and poles of some zeta functions

Abstract

This paper studies zeta functions of the form ${\sum }_{n=1}^{\infty }\chi (n)n^{-s}$, with $\chi$ a completely multiplicative function taking only unimodular values. We denote by $\sigma (\chi )$ the infimum of those $\alpha$ such that the Dirichlet series ${\sum }_{n=1}^{\infty }\chi (n)n^{-s}$ can be continued meromorphically to the half-plane $\mathrm{Re}s>\alpha$, and denote by ${\zeta }_{\chi }(s)$ the corresponding meromorphic function in $\mathrm{Re}s>\sigma (\chi )$. We construct ${\zeta }_{\chi }(s)$ that have $\sigma (\chi )\le 1/2$ and are universal for zero-free analytic functions on the half-critical strip $1/2<\mathrm{Re}s<1$, with zeros and poles at any discrete multisets lying in a strip to the right of $\mathrm{Re}s=1/2$ 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 $\beta +i\gamma$ with $\beta \le 1-\lambda \log \log |\gamma |/\log |\gamma |$ when $\lambda >1$. Finally, we show that there exists ${\zeta }_{\chi }(s)$ with $\sigma (\chi )\le 1/2$ and zeros at any discrete multiset in the strip $1/2<\mathrm{Re}s\le 39/40$ with no accumulation point in $\mathrm{Re}s>1/2$; on the Riemann hypothesis, this strip may be replaced by the half-critical strip $1/2<\mathrm{Re}s<1$.

This is the final version of the paper which has been accepted for publication in Journal d'Analyse Mathématique