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2012 | OriginalPaper | Buchkapitel

The heat kernel, theta inversion and zetas on Г∖G∕K

verfasst von : Jay Jorgenson, Serge Lang

Erschienen in: Number Theory, Analysis and Geometry

Verlag: Springer US

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Abstract

Direct and precise connections between zeta functions with functional equations and theta functions with inversion formulas can be made using various integral transforms, namely Laplace, Gauss, and Mellin transforms as well as their inversions. In this article, we will describe how one can initiate the process of constructing geometrically defined zeta functions by beginning inversion formulas which come from heat kernels. We state conjectured spectral expansions for the heat kernel, based on the so-called heat Eisenstein series defined in [JoL 04]. We speculate further, in vague terms, the goal of constructing a type of ladder of zeta functions and describe similar features from elsewhere in mathematics.

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Vorheriges Kapitel Uniform estimates for primitive divisors in elliptic divisibility sequences

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

The present article was completed by Jorgenson and Lang during the summer of 2005 shortly before Lang passed away on September 12, 2005. As such, this article is the last mathematics paper written by Lang. At the time the article was written, it was the intention to describe the future direction that Jorgenson and Lang envisioned for their research.

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Titel: The heat kernel, theta inversion and zetas on Г∖G∕K
verfasst von: Jay Jorgenson
Serge Lang
Verlag: Springer US
Buch: Number Theory, Analysis and Geometry
Print ISBN: 978-1-4614-1259-5

Electronic ISBN: 978-1-4614-1260-1

Copyright-Jahr: 2012
DOI: https://doi.org/10.1007/978-1-4614-1260-1_13

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