English

Universality of the Hurwitz zeta-function on the half plane of absolute convergence

Number Theory 2020-08-12 v1 Complex Variables

Abstract

Let KK be a compact set with connected complement on the half-plane Re(s)>0(s)>0, and let ff be a continuous function on KK which is analytic in its interior. We prove that for any parameter 0<α<1,α120<\alpha<1, \alpha \neq \frac 1 2 then f(s)f(s) may be uniformly approximated arbitrarily closely by ζ(1+iT+iδs,α)\zeta(1+iT+i\delta s,\alpha) on KK for some T,δ>0T,\delta>0, where ζ(s,α)\zeta(s,\alpha) denote the Hurwitz zeta-function. This is the first known universality result that is also known to hold for the Hurwitz zeta-function with an algebraic irrational parameter.

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Cite

@article{arxiv.2008.04709,
  title  = {Universality of the Hurwitz zeta-function on the half plane of absolute convergence},
  author = {Johan Andersson},
  journal= {arXiv preprint arXiv:2008.04709},
  year   = {2020}
}

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16 pages