Identities involving Farey fractions

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.