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arXiv · 2405.00274

Large Values of Newform Dedekind Sums

Georgia Corbett·Matthew P. Young
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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.

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Georgia Corbett, Matthew P. Young. 2024-05-01. Large Values of Newform Dedekind Sums. https://arxiv.org/abs/2405.00274

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