Relations between theta-functions, Hardy sums, Eisenstein and Lambert series in the transformation formula of log \(\eta_{g,h}(z)\).
The author defines logarithms of generalized theta functions \(f_{n;g,h}(z, N)\) for \(n= 2,3,4\) from generalized Dedekind eta functions \(\eta_{g,h}(z, N)\), analogously to the expressions of classical theta functions \(\vartheta_n(z)\) for \(n= 2,3,4\) in terms of the Dedekind eta function \(\eta(z)\). When \((g, h)= (0, 0)\), \(f_{n;g,h}(z, N)\) coincides with \(2\log\vartheta_n(z)\). Then first, the transformation formula of \(f_{n;g,h}(z, N)\) under the appropriate congruence subgroup of \(\text{SL}_2(\mathbb{Z})\) is derived for each \(n\), where sums similar to Hardy sums appear. Secondly, the author considers the first derivative of \(f_{n;g,h}(z, N)\) with respect to \(z\) and finds the expression of it in terms of generalized Eisenstein series. Finally, the author considers Lambert series, which are connected with generalized Hardy sums. Here generalizations are given by the replacement of classical Dedekind sums with Apostol's generalized Dedekind sums.
scientific article; zbMATH DE number 1915510
scientific article