Author:
Locus Dawsey Madeline,Masri Riad
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.
Subject
Applied Mathematics,General Mathematics
Cited by
18 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献