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Modular relations and parity in number theory

Record Type:	Electronic resources : Monograph/item
Title/Author:	Modular relations and parity in number theoryby Kalyan Chakraborty, Shigeru Kanemitsu, Takako Kuzumaki.
Author:	Chakraborty, Kalyan.
other author:	Kanemitsu, Shigeru.
Published:	Singapore :Springer Nature Singapore :2025.
Description:	xii, 267 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Number theory.
Online resource:	https://doi.org/10.1007/978-981-96-6471-9
ISBN:	9789819664719$q(electronic bk.)

Modular relations and parity in number theory
Chakraborty, Kalyan.

Modular relations and parity in number theory[electronic resource] /by Kalyan Chakraborty, Shigeru Kanemitsu, Takako Kuzumaki. - Singapore :Springer Nature Singapore :2025. - xii, 267 p. :ill., digital ;24 cm. - Infosys Science Foundation series in mathematical sciences,2364-4044. - Infosys Science Foundation series in mathematical sciences..

Introduction and preliminaries -- Ramified functional equations and (CSIF) -- Unified theory of Epstein zeta-functions & zeta-functions associated with real-analytic automorphic forms -- A unifying principle for modular relations and summation formulas -- Class numbers of Abelian fields and (CPMD) -- Historical remarks.

This book describes research problems by unifying and generalizing some remote-looking objects through the functional equation and the parity relation of relevant zeta functions, known as the modular relation or RHB correspondence. It provides examples of zeta functions introduced as absolutely convergent Dirichlet series, not necessarily with the Euler product. The book generalizes this to broader cases, explaining the special functions involved. The extension of the Chowla-Selberg integral formula and the Hardy transform are key, substituting the Bochner modular relation in the zeta function of Maass forms. The book also develops principles to deduce summation formulas as modular relations and addresses Chowla's problem and determinant expressions for class numbers. Many books define zeta functions using Euler products, excluding Epstein and Hurwitz-type zeta functions. Euler products are constructed from objects with a unique factorization domain property. This book focuses on using the functional equation, called the modular relation, specifically the ramified functional equation of the Hecker type. Here, the gamma factor is the product of two gamma functions, leading to the Fourier-Whittaker expansion, and reducing to the Fourier-Bessel expansion or the Chowla-Selberg integral formula for Epstein zeta functions.

ISBN: 9789819664719$q(electronic bk.)

Standard No.: 10.1007/978-981-96-6471-9doiSubjects--Topical Terms:

189521
Number theory.

LC Class. No.: QA241

Dewey Class. No.: 512.7

Modular relations and parity in number theory
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520 $a This book describes research problems by unifying and generalizing some remote-looking objects through the functional equation and the parity relation of relevant zeta functions, known as the modular relation or RHB correspondence. It provides examples of zeta functions introduced as absolutely convergent Dirichlet series, not necessarily with the Euler product. The book generalizes this to broader cases, explaining the special functions involved. The extension of the Chowla-Selberg integral formula and the Hardy transform are key, substituting the Bochner modular relation in the zeta function of Maass forms. The book also develops principles to deduce summation formulas as modular relations and addresses Chowla's problem and determinant expressions for class numbers. Many books define zeta functions using Euler products, excluding Epstein and Hurwitz-type zeta functions. Euler products are constructed from objects with a unique factorization domain property. This book focuses on using the functional equation, called the modular relation, specifically the ramified functional equation of the Hecker type. Here, the gamma factor is the product of two gamma functions, leading to the Fourier-Whittaker expansion, and reducing to the Fourier-Bessel expansion or the Chowla-Selberg integral formula for Epstein zeta functions.
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