Electronic Communications in Probability

A Note on the $S_2(\delta)$ Distribution and the Riemann Xi Function

Dmitry Ostrovsky

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Abstract

The theory of $S_2(\delta)$ family of probability distributions is used to give a derivation of the functional equation of the Riemann xi function. The $\delta$ deformation of the xi function is formulated in terms of the $S_2(\delta)$ distribution and shown to satisfy Riemann's functional equation.  Criteria for simplicity of roots of the xi function and for its simple roots to satisfy the Riemann hypothesis are formulated in terms of a differentiability property of the $S_2(\delta)$ family. For application, the values of the Riemann zeta function at the integers and of the Riemann xi function in the complex plane are represented as integrals involving the Laplace transform of $S_2.$

Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 85, 13 pp.

Dates
Accepted: 11 December 2014
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465316787

Digital Object Identifier
doi:10.1214/ECP.v19-3608

Mathematical Reviews number (MathSciNet)
MR3291622

Zentralblatt MATH identifier
1352.11076

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 30D10: Representations of entire functions by series and integrals 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms

Keywords
Riemann xi function Mellin transform Laplace transform Functional equation Infinite divisibility

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Ostrovsky, Dmitry. A Note on the $S_2(\delta)$ Distribution and the Riemann Xi Function. Electron. Commun. Probab. 19 (2014), paper no. 85, 13 pp. doi:10.1214/ECP.v19-3608. https://projecteuclid.org/euclid.ecp/1465316787


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