Large Values of Newform Dedekind Sums

Abstract

We study a generalized Dedekind sum $S_{\ddot{I}{‡}_1,\ddot{I}{‡}_2}(a,c)$ attached to newform Eisenstein series $E_{\ddot{I}{‡}_1,\ddot{I}{‡}_2}(z,s)$. 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.

9 pages