Improved asymptotics for moments of reciprocal sums for partitions into distinct parts

TL;DR

This paper significantly advances the asymptotic understanding of sums of reciprocals and their squares over partitions into distinct parts, overcoming challenges posed by non-modular generating functions.

Contribution

It provides improved asymptotic formulas for reciprocal sums in partitions into distinct parts, using novel methods beyond classical modular form techniques.

Findings

Enhanced asymptotic formulas for $s_1(n)$ and $s_2(n)$

Methods applicable to non-modular generating functions

Deeper understanding of partition reciprocal sums

Abstract

In this paper we strongly improve asymptotics for $s_1(n)$ (respectively $s_2(n)$) which sums reciprocals (respectively squares of reciprocals) of parts throughout all the partitions of $n$ into distinct parts. The methods required are much more involved than in the case of usual partitions since the generating functions are not modular and also do not posses product expansions.