On Solutions to Riemann's Functional Equation

Abstract

In 1921 Hamburger proved that Riemann's functional equation characterizes the Riemann zeta function in the space of functions representable by ordinary Dirichlet series satisfying certain regularity conditions. We consider solutions to a more general functional equation with real weight k. In the case of Hamburger's theorem, k = $- \frac{1}{2}$. We show that, under suitable conditions, the generalized functional equation admits no nontrivial solutions for k > 0 unless k = $- \frac{1}{2}$. Our proof generalizes an elegant proof of Hamburger's theorem given by Siegel, and employs a generalized integral transform.