An Arithmetic Sum Associated with the Classical Theta Function

TL;DR

The paper investigates arithmetic sums S(h,k) and S(k) that arise in the modular transformation of the classical theta function, along with a related sum T(k). It provides explicit formulas for T(k) in terms of divisor sums and Euler totients, derives an asymptotic for a related partial sum via Dirichlet-series methods, and establishes several elementary lower bounds for S(k). Numerical data motivate conjectures about the growth and positivity of S(k) for primes, while the work highlights deep connections to Dedekind-sum analogues and modular hyperbola distributions. The combination of transformation theory, analytic techniques, and elementary number theory advances understanding of these arithmetical sums and their role in theta-transformations.

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.

Figures (2)

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Theorems & Definitions (45)

• Conjecture 1.1
• Conjecture 3.1
• Conjecture 3.2
• Conjecture 3.3
• Lemma 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4