Large Values of Newform Dedekind Sums
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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2024
- DOI:
- 10.48550/arXiv.2405.00274
- arXiv:
- arXiv:2405.00274
- Bibcode:
- 2024arXiv240500274C
- Keywords:
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- Mathematics - Number Theory;
- 11F20
- E-Print:
- 9 pages