Higher Hickerson formula

TL;DR

This paper generalizes Hickerson's explicit formula for Dedekind sums to a broader class associated with Todd power series, providing new expressions for partial zeta values and exploring equidistribution properties.

Contribution

It develops a generalized Hickerson formula for Dedekind sums linked to Todd series, extending previous results and applications to zeta values and distribution analysis.

Findings

Derived a generalized explicit formula for $s_{i,j}(p,q)$

Expressed partial zeta values using only the integral part of sums

Analyzed equidistribution of fractional parts in a new context

Abstract

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 $i+j$.