An Arithmetic Sum Associated with the Classical Theta Function

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

An Arithmetic Sum Associated with the Classical Theta Function

Number Theory 2026-01-12 v3

Authors: Bruce C. Berndt , Raghavendra N. Bhat , Jeffrey L. Meyer , Likun Xie , Alexandru Zaharescu

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.

Keywords

divisor function zeta values and euler sums exponential sums

Cite

@article{arxiv.2501.03234,
  title  = {An Arithmetic Sum Associated with the Classical Theta Function},
  author = {Bruce C. Berndt and Raghavendra N. Bhat and Jeffrey L. Meyer and Likun Xie and Alexandru Zaharescu},
  journal= {arXiv preprint arXiv:2501.03234},
  year   = {2026}
}

An Arithmetic Sum Associated with the Classical Theta Function

Abstract

Keywords

Cite

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