A Jentzsch-Theorem for Kapteyn, Neumann, and General Dirichlet Series

TL;DR

This paper extends the classical Jentzsch theorem to Kapteyn, Neumann, and general Dirichlet series, showing that their zeros cluster near the boundary of the domain of convergence, with implications for the distribution of zeros.

Contribution

It proves a Jentzsch-type theorem for a broad class of series including Kapteyn, Neumann, and general Dirichlet series, unifying their zero distribution properties.

Findings

Zeros cluster near the boundary of the convergence domain.

The growth function relates to the convergence abscissa of Dirichlet series.

Zeros near the real axis follow an asymptotic linear density.

Abstract

Comparing phase plots of truncated series solutions of Kepler's equation by Lagrange's power series with those by Bessel's Kapteyn series strongly suggest that a Jentzsch-type theorem holds true not only for the former but also for the latter series: each point of the boundary of the domain of convergence in the complex plane is a cluster point of zeros of sections of the series. We prove this result by studying properties of the growth function of a sequence of entire functions. For series, this growth function is computable in terms of the convergence abscissa of an associated general Dirichlet series. The proof then extends, besides including Jentzsch's classical result for power series, to general Dirichlet series, to Kapteyn, and to Neumann series of Bessel functions. Moreover, sections of Kapteyn and Neumann series generally exhibit zeros close to the real axis which can be explained, including their asymptotic linear density, by the theory of the distribution of zeros of entire functions.