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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2012, Volume 276, Pages 131–145
(Mi tm3374)
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This article is cited in 1 scientific paper (total in 1 paper)
Identities involving Farey fractions
M. N. Huxley School of Mathematics, University of Cardiff, Cardiff, Wales, UK
Abstract:
The rational numbers $a/q$ in $[0,1]$ can be counted by increasing height $H(a/q)=\max(a,q)$, or ordered as real numbers. Franel's identity shows that the Riemann hypothesis is equivalent to a strong bound for a measure of the independence of these two orderings. We give a proof using Dedekind sums that allows weights $w(q)$. Taking $w(q)=\chi(q)$ we find an extension to Dirichlet L-functions.
Received in September 2011
Citation:
M. N. Huxley, “Identities involving Farey fractions”, Number theory, algebra, and analysis, Collected papers. Dedicated to Professor Anatolii Alekseevich Karatsuba on the occasion of his 75th birthday, Trudy Mat. Inst. Steklova, 276, MAIK Nauka/Interperiodica, Moscow, 2012, 131–145; Proc. Steklov Inst. Math., 276 (2012), 125–139
Linking options:
https://www.mathnet.ru/eng/tm3374 https://www.mathnet.ru/eng/tm/v276/p131
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Abstract page: | 228 | Full-text PDF : | 95 | References: | 40 |
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