A new integral equation and integrals associated with number theory

Alexander E. Patkowski

Type: Article

Publication Date: 2021-04-10

Citations: 1

DOI: https://doi.org/10.1007/s10476-021-0076-8

Abstract

We utilize a combination of integral transforms, including the Laplace transform, with some classical results in analytic number theory concerning the Riemann ξ-function, to obtain a new integral equation. This integral equation is generalized to self-dual principal automorphic L-functions. We also provide a new proof of known functional-type identities from analytic number theory, and recast some criteria associated with the RH. An application of our integral equation to the Dirichlet problem in the half plane is stated, giving a new application of the Riemann ξ-function integral.

Locations

arXiv (Cornell University) - View - PDF
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Analysis Mathematica - View

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Works Cited by This (15)

Action	Title	Year	Authors
+	<i>Partial Differential Equations for Scientists and Engineers</i>	1985	Stanley J. Farlow Stephen F. Becker
+ PDF Chat	Turán inequalities and zeros of Dirichlet series associated with certain cusp forms	1994	J. Brian Conrey Amit Ghosh
+ PDF Chat	Character analogues of Ramanujan-type integrals involving the Riemann Ξ-function	2012	Atul Dixit
+	Integral representations for the Dirichlet L-functions and their expansions in Meixner–Pollaczek polynomials and rising factorials†	2007	Alexey Kuznetsov
+ PDF Chat	Expansion of the Riemann Ξ Function in Meixner–Pollaczek Polynomials	2008	Alexey Kuznetsov
+	Series transformations and integrals involving the Riemann Ξ-function	2010	Atul Dixit
+	The integral representation for the Riemann Ξ-function	1971	Robert Spira
+ PDF Chat	Computational strategies for the Riemann zeta function	2000	Jonathan M. Borwein David M. Bradley Richard E. Crandall
+	Fourier Transforms of Positive Definite Kernels and the Riemann $$\xi $$ ξ -Function	2015	George Csordás
+	The Theory of the Riemann Zeta-Function	1987	E. C. Titchmarsh D. R. Heath‐Brown