English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

This paper studies zeta functions of the form $\sum_{n=1}^{\infty} \chi(n)n^{-s}$, with $\chi$ a completely multiplicative function taking onlyunimodular 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 becontinued meromorphically to the half-plane $\operatorname{Re} s>\alpha$, anddenote by $\zeta_{\chi}(s)$ the corresponding meromorphic function in$\operatorname{Re} s>\sigma(\chi)$. We construct $\zeta_{\chi}(s)$ that have$\sigma(\chi)\le 1/2$ and are universal for zero-free analytic functions on thehalf-critical strip $1/2<\operatorname{Re} s <1$, with zeros and poles at anydiscrete multisets lying in a strip to the right of $\operatorname{Re} s =1/2$and satisfying a density condition that is somewhat stricter than the densityhypothesis for the zeros of the Riemann zeta function. On a conceivable versionof Cram\'{e}r's conjecture for gaps between primes, the density condition canbe 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 $\sigma(\chi) \le 1/2$ andzeros at any discrete multiset in the strip $1/2<\operatorname{Re} s \le 39/40$with no accumulation point in $\operatorname{Re} s >1/2$; on the Riemannhypothesis, this strip may be replaced by the half-critical strip $1/2 <\operatorname{Re} s < 1$.

Universality and distribution of zeros and poles of some zeta functions