Effective estimates for the smallest parts function

TL;DR

This paper improves the error term in the asymptotic formula for the smallest parts function by employing explicit bounds on sums of Kloosterman sums of half integral weight, advancing understanding in partition theory.

Contribution

It introduces a new method to tighten error bounds in the asymptotic analysis of the smallest parts function using bounds on Kloosterman sums.

Findings

Significantly improved error term in spt(n) asymptotics

Established explicit bounds for sums of half-integral weight Kloosterman sums

Enhanced analytical techniques for partition-related functions

Abstract

We give a substantial improvement for the error term in the asymptotic formula for the smallest parts function $\mathrm{spt}(n)$ of Andrews. Our methods depend on an explicit bound for sums of Kloosterman sums of half integral weight on the full modular group.