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Applications of heat kernels on abelian groups: ζ(2n), quadratic reciprocity, Bessel integrals

In: Number Theory, Analysis and Geometry

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  • Anders Karlsson

    (Université de Genéve, Section de Mathématiques)

Abstract

The discussion centers around three applications of heat kernel considerations on $$\mathbb{R}$$ , $$\mathbb{Z}$$ and their quotients. These are Euler’s formula for ζ(2n), Gauss’ quadratic reciprocity law, and the evaluation of certain integrals of Bessel functions. Some further applications are mentioned, including the functional equation of Riemann’s ζ-function, the reflection formula for the Γ-function, and certain infinite sums of Bessel functions.

Suggested Citation

  • Anders Karlsson, 2012. "Applications of heat kernels on abelian groups: ζ(2n), quadratic reciprocity, Bessel integrals," Springer Books, in: Dorian Goldfeld & Jay Jorgenson & Peter Jones & Dinakar Ramakrishnan & Kenneth Ribet & John Tate (ed.), Number Theory, Analysis and Geometry, edition 127, pages 307-320, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-1260-1_14
    DOI: 10.1007/978-1-4614-1260-1_14
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