Riesz summability on boundary lines of holomorphic functions generated by Dirichlet series

TL;DR

This paper investigates the boundary behavior of holomorphic functions generated by Dirichlet series, establishing criteria for their Riesz summability on boundary lines, extending classical convergence results to more general settings.

Contribution

It introduces new criteria for Riesz summability of Dirichlet series boundary functions, generalizing classical convergence theorems to a broader class of functions.

Findings

Criteria for pointwise Riesz summability on boundary lines.

Conditions for uniform Riesz summability.

Extension of classical convergence principles to Dirichlet series.

Abstract

A particular consequence of the famous Carleson-Hunt theorem is that the Taylor series expansions of bounded holomorphic functions on the open unit disk converge almost everywhere on the boundary, whereas on single points the convergence may fail. In contrast, Bayart, Konyagin, and Queff\'elec constructed an example of an ordinary Dirichlet series $\sum a_n n^{-s}$, which on the open right half-plane $[Re >0]$ converges pointwise to a bounded, holomorphic function -- but diverges at each point of the imaginary line, although its limit function extends continuously to the closed right half plane. Inspired by a result of M.~Riesz, we study the boundary behavior of holomorphic functions $f$ on the right half-plane which for some $\ell \ge 0$ satisfy the growth condition $|f(s)| = O((1 + |s|)^\ell)$ and are generated by some Riesz germ, i.e., there is a frequency $\lambda = (\lambda_n)$ and a $\lambda$-Dirichlet series $\sum a_n e^{-\lambda_n s}$ such that on some open subset of $[Re >0]$ and for some $m \ge 0$ the function $f$ coincides with the pointwise limit (as $x \to \infty$) of so-called $(\lambda,m)$-Riesz means $\sum_{\lambda_n < x} a_n e^{-\lambda_n s}\big( 1-\frac{\lambda_n}{x}\big)^m ,\,x >0\,.$ Our main results present criteria for pointwise and uniform Riesz summability of such functions on the boundary line $[Re =0]$, which includes conditions that are motivated by classics like the Dini-test or the principle of localization.