Let k ≥ 1 denote any positive rational integer. We give formulae for the sums $S_{odd}(k,f)={\sum }_{\chi (-1)=-1}\left|L(k,\chi )\right|²$ (where χ ranges over the ϕ(f)/2 odd Dirichlet characters modulo f > 2) whenever k ≥ 1 is odd, and for the sums $S_{even}(k,f)={\sum }_{\chi (-1)=+1}\left|L(k,\chi )\right|²$ (where χ ranges over the ϕ(f)/2 even Dirichlet characters modulo f>2) whenever k ≥ 1 is even.

We study two rather different problems, one arising from Diophantine geometry and one arising from Fourier analysis, which lead to very similar questions, namely to the study of the ranks of matrices with entries either zero or $\left(\right(xy/q\left)\right),(0\le x,y\<q)$, where $\left(\right(u\left)\right)=u-\left[u\right]-1/2$ denotes the “centered” fractional part of $x$. These ranks, in turn, are closely connected with the non-vanishing of the Dirichlet $L$-functions at $s=1$.