Aspects of Zeta-Function Theory in the Mathematical Works of Adolf Hurwitz

TL;DR

This paper explores Adolf Hurwitz's early and ongoing contributions to zeta-function theory, highlighting his pioneering insights into properties of the Epstein and Hurwitz zeta-functions well before their formal development.

Contribution

It reveals Hurwitz's early awareness and study of zeta-functions, emphasizing his role in foundational developments prior to Epstein and others.

Findings

Hurwitz understood the analytic properties of the Epstein zeta-function in 1889.

Hurwitz's diaries show a lifelong engagement with zeta-function theory.

His early work predates and influences subsequent research in the field.

Abstract

Adolf Hurwitz is rather famous for his celebrated contributions to Riemann surfaces, modular forms, diophantine equations and approximation as well as to certain aspects of algebra. His early work on an important generalization of Dirichlet's $L$-series, nowadays called Hurwitz zeta-function, is the only published work settled in the very active field of research around the Riemann zeta-function and its relatives. His mathematical diaries, however, provide another picture, namely a lifelong interest in the development of zeta-function theory. In this note we shall investigate his early work, its origin and its reception, as well as Hurwitz's further studies of the Riemann zeta-function and allied Dirichlet series from his diaries. It turns out that Hurwitz already in 1889 knew about the essential analytic properties of the Epstein zeta-function (including its functional equation) 13 years before Paul Epstein.