Alexander Ostrowski's "On Dirichlet Series and Algebraic Differential Equations"

This is an English translation of Ostrowski’s article “Über 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.