Dedekind sums and Fourier coefficients of modular forms
The result may be considered as a reasonably final answer for a sufficiently old (1972) question of H. Rademacher and E. Grosswald. Let for \(c\geq 1\) and \((d,c)=1\), S(d,c) be the Dedekind sum \(S(d,c)=\sum^{c-1}_{m=1}\{md/c\}\{m/c\}\), \(\{x\}=x-[x]-1/2\) if \(x\not\in {\mathbb{Z}}\) and \(\{x\}=0\) for \(x\in {\mathbb{Z}}\). Then for \(M\to +\infty\) \[ \frac{\log M}{M}\sum^{M}_{c=1}\frac{1}{c}\sum_{(d,c)=1}g(S(d,c),\frac{d}{c})=\frac{12}{\pi^ 2}\int^{\infty}_{- \infty}\int_{{\mathbb{R}}/{\mathbb{Z}}}g(x,y) dy dx+o(1) \] for each compactly supported continuous function g on \({\mathbb{R}}\times ({\mathbb{R}}/{\mathbb{Z}})\).
scientific article; zbMATH DE number 4191779