On a property of elliptic Dedekind sums
Let L be a lattice in \({\mathbb{C}}\) with multiplier ring \({\mathcal O}=\{m\in {\mathbb{C}}|\) mL\(\subset L\}\). Let \(E_ k(x)\) be the value of the continuation of \(\sum_{w\in L}(w+x)^{-k} | w+x|^{-s}\) at \(s=0\). \textit{R. Sczech} [Invent. Math. 76, 523-551 (1984; Zbl 0521.10021)] introduced the sums \[ D(a,c)=(1/c)\sum_{k\in L/cL}E_ 1(ak/c)E_ 1(k/c) \] for a, c in \({\mathcal O}\) with \(c\neq 0\), an analog of the classical Dedekind sums, and conjectured that \(D(a,c)=-D(\bar a,\bar c).\) This formula can be derived from previous work of the author [J. Reine Angew. Math. 373, 148-165 (1987; Zbl 0601.10021)]. Now the author gives a short proof of this formula which rests upon a well-known functional equation of an Eisenstein series. He also discusses some consequences.
scientific article; zbMATH DE number 4012720