Large Values of Newform Dedekind Sums

TL;DR

This paper investigates the size of generalized Dedekind sums associated with Eisenstein series, revealing they are typically bounded by a logarithmic cubic function and providing examples of large values through continued fraction analysis.

Contribution

It establishes bounds on the size of Dedekind sums linked to newform Eisenstein series and connects their magnitude to continued fraction properties, complemented by explicit computations.

Findings

Dedekind sums are rarely larger than log^3 c

Continued fractions control the size of Dedekind sums

Explicit examples of large Dedekind sums are provided

Abstract

We study a generalized Dedekind sum $S_{\chi_1,\chi_2}(a,c)$ attached to newform Eisenstein series $E_{\chi_1,\chi_2}(z,s)$. Our work shows the Dedekind sum is rarely substantially larger than $\log^3 c$. The method of proof first relates the size of the Dedekind sum to continued fractions. A result of Hensley from 1991 then controls the average size of the maximal partial quotient in the continued fraction expansion of $a/c$. We complement this result by computing approximate values of the Dedekind sum in some special cases, which in particular produces examples of large values of the Dedekind sum.