Geometric Aspects of Quasi-Periodic Property of Dirichlet Functions

Abstract

The concept of quasi-periodic property of a function has been introduced by Harald Bohr in 1921 and it roughly means that the function comes (quasi)-periodically as close as we want on every vertical line to the value taken by it at any point belonging to that line and a bounded domain Ω . He proved that the functions defined by ordinary Dirichlet series are quasi-periodic in their half plane of uniform convergence. We realized that the existence of the domain Ω is not necessary and that the quasi-periodicity is related to the denseness property of those functions which we have studied in a previous paper. Hence, the purpose of our research was to prove these two facts. We succeeded to fulfill this task and more. Namely, we dealt with the quasi-periodicity of general Dirichlet series by using geometric tools perfected by us in a series of previous projects. The concept has been applied to the whole complex plane (not only to the half plane of uniform convergence) for series which can be continued to meromorphic functions in that plane. The question arise: in what conditions such a continuation is possible? There are known examples of Dirichlet series which cannot be continued across the convergence line, yet there are no simple conditions under which such a continuation is possible. We succeeded to find a very natural one.

On the Non-Trivial Zeros of Dirichlet Functions

“…(1) converges uniformly on compact sets included in the half plane Re c s σ > and therefore it is an analytic function there. The abscissa of absolute convergence a σ of the series ( 1) is by definition the abscissa of convergence of the se- has been studied by Cahen [1] who proved (see also [12]): Proposition 1: If the series (1) has a finite abscissa of convergence, then the series (4) has the abscissa of convergence zero.…”

“…The sufficient condition found in [12] for the series (1) to admit meromorphic continuation into the whole complex plane can be relaxed as follows:…”

“…The limit (8) guarantees that this integral is convergent no matter if ( ) is easy to check that these series have both the imaginary axis as natural boundary, hence they cannot be meromorphically continued across the imaginary axis. We can give a direct proof for the series (12), as an easy exercise. The Hadamard's formula shows that the radius of convergence of this series is 1, i.e.…”

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

“…They are implemented in Mathematica and some affirmations about general Dirichlet functions are illustrated by using Dirichlet L-functions. However, the interest in more general functions is obvious and we have recently devoted to them a lot of publications (see [2]- [15]). An account of recent advances in this field can be found in [8].…”

“…for the extended function when it exists and we call it Dirichlet function. Following Speiser [16], who studied the Riemann Zeta function, we have used in [2]- [15] the pre-image of the real axis by We have proved (see for example [8] In the case of a strip , 0 when is the case, as in Figure 2). These are strips unbounded to the right and to the left when boundaries of fundamental domains bounded to the right.…”