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 $n\ge N(\epsilon )$, we have \begin{equation*} \frac{\sqrt{6}}π\sqrt{n}\,p(n)<\mathrm{spt}(n)<\left(\frac{\sqrt{6}}π+ε\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.

arXiv commentChanged the title. Added more details and simplified some arguments in Section 3