Electronic Communications in Probability

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

Dmitry Ostrovsky

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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

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

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

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Zentralblatt MATH identifier

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

Riemann xi function Mellin transform Laplace transform Functional equation Infinite divisibility

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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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