Higher Hickerson formula

• Published in: arXiv.org
• Main Authors: Lee, Jungyun, Byungheup Jun, Chae, Hi-joon
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 08.08.2016
• Subjects: Integrals; Power series; Sums
• ISSN: 2331-8422

Hickerson made an explicit formula for Dedekind sums \(s(p,q)\) in terms of the continued fraction of \(p/q\). We develop analogous formula for generalized Dedekind sums \(s_{i,j}(p,q)\) defined in association with the \(x^{i}y^{j}\)-coefficient of the Todd power series of the lattice cone in \(\Bbb{R}^2\) generated by \((1,0)\) and \((p,q)\). The formula generalizes Hickerson's original one and reduces to Hickerson's for \(i=j=1\). In the formula, generalized Dedekind sums are divided into two parts: the integral \(s^I_{ij}(p,q)\) and the fractional \(s^R_{ij}(p,q)\). We apply the formula to Siegel's formula for partial zeta values at a negative integer and obtain a new expression which involves only \(s^I_{ij}(p,q)\) the integral part of generalized Dedekind sums. This formula directly generalize Meyer's formula for the special value at \(0\). Using our formula, we present the table of the partial zeta value at \(s=-1\) and \(-2\) in more explicit form. Finally, we present another application on the equidistribution property of the fractional parts of the graph \(\Big{(}\frac{p}{q},R_{i+j}q^{i+j-2} s_{ij}(p,q)\Big{)}\) for a certain integer \(R_{i+j}\) depending on \(i+j\).