On logarithmic integrals of the Riemann zeta-function and an approach to the Riemann Hypothesis by a geometric mean with respect to an ergodic transformation

摘要

We study the distribution of certain improper integrals associated with the Riemann zeta-function (zeta ), namely, where (a>0) is real. For instance, for (a=sigma in (0, 1]), we shall show that In particular, for (sigma = 1/2), we get a well-known result proved by Balazard, Saias and Yor: In the final section, we study Boole’s transformation, (T_a:x mapsto (x - a^2/x)/2) for (xne 0) and (T_a0=0), and show by its ergodicity that, for (sigma in mathbb {R}), the geometric mean-value of exists for almost all (xin mathbb {R}) as (n rightarrow + infty ), and is independent of x. In particular, for (sigma = 1/2), we obtain a criterion for the Riemann Hypothesis: let for fixed (n in mathbb {N}) and (mu in [0,pi ]), then a necessary condition for the truth of the Riemann Hypothesis is with arbitrary (alpha in mathbb {R}). Keywords Riemann zeta-function Riemann Hypothesis Ergodic theory Logarithmic integrals Cauchy distributed function Cauchy random walk Entire functions Mathematics Subject Classification 11M26 11K06 11K31 37A45 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (22) References1.Adler, R.L., Weiss, B.: The ergodic infinite measure preserving transformation of Boole. Israel J. Math. 16(3), 263–278 (1973)MathSciNetCrossRef2.Aleman, A., Feldman, N.S., Ross, W.T.: The Hardy Space of a Slit Domain. Frontiers in Mathematics. Birkhäuser, Basel (2009)CrossRef3.Applebaum, D.: Lévy Processes and Stochastic Calculus. 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Acta Arith. 46(4), 369–395 (1986)MATHMathSciNet About this Article Title On logarithmic integrals of the Riemann zeta-function and an approach to the Riemann Hypothesis by a geometric mean with respect to an ergodic transformation Journal European Journal of Mathematics Volume 1, Issue 4 , pp 829-847 Cover Date2015-12 DOI 10.1007/s40879-015-0073-1 Print ISSN 2199-675X Online ISSN 2199-6768 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Algebraic Geometry Keywords Riemann zeta-function Riemann Hypothesis Ergodic theory Logarithmic integrals Cauchy distributed function Cauchy random walk Entire functions 11M26 11K06 11K31 37A45 Authors Lahoucine Elaissaoui (1) Zine El-Abidine Guennoun (1) Author Affiliations 1. Department of Mathematics, Faculty of Sciences, Mohammed V University, 4 Street Ibn Battouta, B.P. 1014 RP, Rabat, Morocco Continue reading... 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