Effective bounds for the Andrews spt-function

Abstract

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 \frac{\sqrt{6}}{\pi}\sqrt{n}\,p(n)<{\mathrm{spt}}(n)<\bigg{(}\frac{\sqrt{6}}{% \pi}+\epsilon\bigg{)}\sqrt{n}\,p(n). 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.