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Research Article Open access CC BY 4.0

Geometric Aspects of Denseness Theorems for Dirichlet Functions

Dorin Ghisa, Andrei Horvat-Marc

Journal of Advances in Mathematics and Computer Science · pp. 1–11 · Published 21 Nov 2017

10.9734/JAMCS/2017/37947

Abstract

The first theorem related to the denseness of the image of a vertical line Re s = σ0, σ0 > 1 by the Riemann Zeta function has been proved by Harald Bohr in 1911. We argue that this theorem is not really a denseness theorem. Later Bohr and Courant proved similar theorems for the case 1/2 < Re s ≤ 1. Their results have been generalized to classes of Dirichlet functions and are at the origin of a burgeoning field in analytic number theory, namely the universality theory. The tools used in this theory are mainly of an arithmetic nature and do not allow a visualization of the phenomena involved. Our method is based on conformal mapping theory and is supported by computer generated illustrations. We generalize and refine Bohr and Courant results.

Denseness theorems Diophantine approximation general Dirichlet series fundamental domains.

References (3)

  1. 1 Fundamental Domains and Analytic Continuation of General Dirichlet Series [DOI]
  2. 2 On the generalized Riemann hypothesis II [DOI]
  3. 3 The Geometry of the Mappings by General Dirichlet Series [DOI]

Cited by 4

The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

D. Ghisa · Advances in Pure Mathematics · 2020

Geometric Aspects of Quasi-Periodic Property of Dirichlet Functions

D. Ghisa, Andrei Horvat-Marc · 2018

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