Asymptotic formulas for the coefficients of the truncated theta series

TL;DR

This paper derives asymptotic formulas for coefficients of truncated theta series and confirms Merca's conjectures on their non-negativity for large N, advancing understanding of these series.

Contribution

It introduces Wright's Circle Method to analyze truncated theta series and proves Merca's conjectures for sufficiently large N.

Findings

Asymptotic formulas for coefficients derived

Merca's conjectures confirmed for large N

Enhanced understanding of truncated theta series

Abstract

Motivated by the groundbreaking work of Andrews and Merca, truncated theta series have been extensively studied over the years. In particular, Merca made conjectures on the non-negativity of the coefficient of $q^N$ in truncated series from the Jacobi triple product identity and the quintuple product identity. In this paper, using Wright's Circle Method, we establish asymptotic formulas for the coefficients of truncated theta series and prove that Merca's conjectures are true for sufficiently large $N$.