Computational Aspects of Hamburger’s Theorem

Abstract

Riemann’s zeta function (defined by a certain Dirichlet series) satisfies an identity known as the functional equation. H. Hamburger established that the function is identified by the equation inside a wide class of functions defined by Dirichlet series. Riemann’s zeta function is a member of a large family of functions with similar properties, in particular, satisfying certain functional equations. Hamburger’s theorem can be extended to some (but not to all) of these equations. The paper addresses the following question: how could we discover the Dirichlet series satisfying given functional equation? Two “rules of thumb” for performing such discoveries via numerical computations are demonstrated for functional equations satisfied by Dirichlet eta function, Ramanujan tau L-function, and Davenport–Heilbronn function. A conjectured discrete version of Hamburger’s theorem is stated.