  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.
