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2014 A Note on the $S_2(\delta)$ Distribution and the Riemann Xi Function
Dmitry Ostrovsky
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Electron. Commun. Probab. 19: 1-13 (2014). DOI: 10.1214/ECP.v19-3608


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 $S_2.$


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Dmitry Ostrovsky. "A Note on the $S_2(\delta)$ Distribution and the Riemann Xi Function." Electron. Commun. Probab. 19 1 - 13, 2014.


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

zbMATH: 1352.11076
MathSciNet: MR3291622
Digital Object Identifier: 10.1214/ECP.v19-3608

Primary: 11M06
Secondary: 30D05 , 30D10 , 60E07 , 60E10

Keywords: functional equation , Infinite divisibility , Laplace transform , Mellin transform , Riemann xi function


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