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This is an English translation of Ostrowski's article "\"Uber DirichletscheReihen und algebraische Differentialgleichungen" published German in Math.Zeit. vol. 8, 1920, pp. 241-298. In this article, Ostrowski proves a conjectureof 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 ofone variable and algebraic functions of any number of variables.

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