Effective Bounds for the Andrews spt-function

Abstract

In this paper, we establish an asymptotic formula with an effective bound on the error term for the Andrews smallest parts function $\mathrm{spt}(n)$. We use this formula to prove recent conjectures of Chen concerning inequalities which involve the partition function $p(n)$ and $\mathrm{spt}(n)$. Further, we strengthen one of the conjectures, and prove that for every $\epsilon>0$ there is an effectively computable constant $N(\epsilon) > 0$ such that for all $n\geq N(\epsilon)$, we have \begin{equation*} \frac{\sqrt{6}}{\pi}\sqrt{n}\,p(n)<\mathrm{spt}(n)<\left(\frac{\sqrt{6}}{\pi}+\epsilon\right) \sqrt{n}\,p(n). \end{equation*} Due to the conditional convergence of the Rademacher-type formula for $\mathrm{spt}(n)$, we must employ methods which are completely different from those used by Lehmer to give effective error bounds for $p(n)$. Instead, our approach relies on the fact that $p(n)$ and $\mathrm{spt}(n)$ can be expressed as traces of singular moduli.