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Computational Aspects of Hamburger’s Theorem

  • Yuri MatiyasevichEmail author
Conference paper
Part of the Fields Institute Communications book series (FIC, volume 82)

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.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.St. Petersburg Department of V. A. Steklov Mathematical InstituteSaint PetersburgRussia

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