An Arithmetic Sum Associated with the Classical Theta Function

2024/12/10 by Berndt, Bruce C., Bhat, Raghavendra N., Meyer, Jeffrey L. +2
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.03234

Abstract

The sum S(h,k):=∑_j=1^k-1(-1)^j+1+[hj/k] appears in the modular transformation formulae of the classical theta function ϑ₃(z). The double sum S(k) := ∑_h=1^k-1S(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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