Effective Bounds for the Andrews spt-function

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 $ngeq N(epsilon)$, we have begin{equation*} frac{sqrt{6}}{pi}sqrt{n},p(n)<mathrm{spt}(n)<left(frac{sqrt{6}}{pi}+epsilonright) 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.