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arXiv · 2501.03234

An Arithmetic Sum Associated with the Classical Theta Function

Bruce C. Berndt·Raghavendra N. Bhat·Jeffrey L. Meyer·Likun Xie·Alexandru Zaharescu
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Abstract

The sum $S(h,k):=\sum_{j=1}^{k-1}(-1)^{j+1+[hj/k]}$ appears in the modular transformation formulae of the classical theta function $\vartheta_3(z)$. The double sum $S(k) := \sum_{h=1}^{k-1}S(h,k)$ has a remarkable distribution of values. Although properties for $S(k)$ and a related sum can be established, several interesting conjectures are open.

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Bruce C. Berndt, Raghavendra N. Bhat, Jeffrey L. Meyer, Likun Xie, Alexandru Zaharescu. 2026-01-09. An Arithmetic Sum Associated with the Classical Theta Function. https://arxiv.org/abs/2501.03234

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