An Arithmetic Sum Associated with the Classical Theta Function

TL;DR

This paper investigates a special sum related to the classical theta function, exploring its properties, distribution, and open conjectures, contributing to the understanding of modular transformations in mathematical analysis.

Contribution

It introduces and analyzes the sum $S(h,k)$ and its aggregate $S(k)$, revealing their properties and distribution, and discusses open conjectures in the area.

Findings

Distribution of $S(k)$ values is remarkable

Properties of $S(h,k)$ are established

Several conjectures remain open

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.