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

Abstract

The discussion centers around three applications of heat kernel considerations on, 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.