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Identities involving Farey fractions

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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) = χ(q) we find an extension to Dirichlet L-functions.

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  1. School of Mathematics, University of Cardiff, 23 Senghenydd Road, Cardiff, CF24 4YH, Wales, UK

    M. N. Huxley

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  1. M. N. Huxley
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Correspondence to M. N. Huxley.

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Dedicated to the 75th birthday of A.A. Karatsuba

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Huxley, M.N. Identities involving Farey fractions. Proc. Steklov Inst. Math. 276, 125–139 (2012). https://doi.org/10.1134/S0081543812010105

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