  This is an English translation of Ostrowski's article "\"Uber Dirichletsche
Reihen und algebraische Differentialgleichungen" published German in Math.
Zeit. vol. 8, 1920, pp. 241-298. In this article, Ostrowski proves a conjecture
of Hilbert that the two variable function $\zeta(x,s)=\sum_{n\ge 1}
\frac{x^n}{n^s}$ cannot be written as a composition of analytic functions of
one variable and algebraic functions of any number of variables.
