Geodesic
Identities involving Farey fractions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 131-145

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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. 
@article{TM_2012_276_a9,
     author = {M. N. Huxley},
     title = {Identities involving {Farey} fractions},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {131--145},
     publisher = {mathdoc},
     volume = {276},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2012_276_a9/}
}
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M. N. Huxley. Identities involving Farey fractions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 131-145. http://geodesic.mathdoc.fr/item/TM_2012_276_a9/