Riemann Hypothesis and Random Walks: The Zeta Case

In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its <i>L</i>-function is valid to the right of the critical line <inline-formula><mat...

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Autor principal:	André LeClair
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	zeta functions prime numbers probabilistic number theory Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/5e18b28a5dc940f18d2a12d3425e8e43
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spelling	oai:doaj.org-article:5e18b28a5dc940f18d2a12d3425e8e432021-11-25T19:06:03ZRiemann Hypothesis and Random Walks: The Zeta Case10.3390/sym131120142073-8994https://doaj.org/article/5e18b28a5dc940f18d2a12d3425e8e432021-10-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2014https://doaj.org/toc/2073-8994In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its <i>L</i>-function is valid to the right of the critical line <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℜ</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>></mo><mstyle scriptlevel="0" displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></semantics></math></inline-formula>, and the Riemann hypothesis for this class of <i>L</i>-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet <i>L</i>-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mn>100</mn></msup></semantics></math></inline-formula>-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.André LeClairMDPI AGarticlezeta functionsprime numbersprobabilistic number theoryMathematicsQA1-939ENSymmetry, Vol 13, Iss 2014, p 2014 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	zeta functions prime numbers probabilistic number theory Mathematics QA1-939
spellingShingle	zeta functions prime numbers probabilistic number theory Mathematics QA1-939 André LeClair Riemann Hypothesis and Random Walks: The Zeta Case
description	In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its <i>L</i>-function is valid to the right of the critical line <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℜ</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>></mo><mstyle scriptlevel="0" displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></semantics></math></inline-formula>, and the Riemann hypothesis for this class of <i>L</i>-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet <i>L</i>-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mn>100</mn></msup></semantics></math></inline-formula>-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.
format	article
author	André LeClair
author_facet	André LeClair
author_sort	André LeClair
title	Riemann Hypothesis and Random Walks: The Zeta Case
title_short	Riemann Hypothesis and Random Walks: The Zeta Case
title_full	Riemann Hypothesis and Random Walks: The Zeta Case
title_fullStr	Riemann Hypothesis and Random Walks: The Zeta Case
title_full_unstemmed	Riemann Hypothesis and Random Walks: The Zeta Case
title_sort	riemann hypothesis and random walks: the zeta case
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/5e18b28a5dc940f18d2a12d3425e8e43
work_keys_str_mv	AT andreleclair riemannhypothesisandrandomwalksthezetacase
_version_	1718410280259551232

Riemann Hypothesis and Random Walks: The Zeta Case

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