Vector-valued general Dirichlet series

TL;DR

This paper initiates a systematic study of vector-valued general Dirichlet series, extending classical scalar results to series with coefficients in arbitrary Banach spaces, bridging multiple mathematical disciplines.

Contribution

It introduces the foundational framework for analyzing vector-valued Dirichlet series, expanding the scope of classical theory to Banach space coefficients.

Findings

Establishment of basic properties of vector-valued Dirichlet series

Extension of classical convergence results to Banach space coefficients

Identification of new challenges in the vector-valued setting

Abstract

Opened up by early contributions due to, among others, H. Bohr, Hardy-Riesz, Bohnenblust-Hille, Neder and Landau the last 20 years show a substantial revival of systematic research on ordinary Dirichlet series $\sum a_n n^{-s}$, and more recently even on general Dirichlet series $\sum a_n e^{-\lambda_n s}$. This involves the intertwining of classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. Motivated through this line of research the main goal of this article is to start a systematic study of a variety of fundamental aspects of vector-valued general Dirichlet series $\sum a_n e^{-\lambda_{n} s}$, so Dirichlet series, where the coefficient are not necessarily in $\mathbb{C}$ but in some arbitrary Banach space $X$.