Asymptotics for the partial fractions of the restricted partition generating function I

2015/07/28 by Cormac O'Sullivan, Cormac O’Sullivan, O'Sullivan, Cormac
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #math.NT #msc:11P82 #msc:41A60

paper · pdf · doi:10.48550/arxiv.1507.07975

42 pages

arxiv created 2015/07/28 · arxiv updated 2015/07/30

Abstract

The generating function for p_N(n), the number of partitions of n into at most N parts, may be written as a product of N factors. We find the behavior of coefficients in the partial fraction decomposition of this product as N → ∞ by applying the saddle-point method, where the saddle-point we need is associated to a zero of the analytically continued dilogarithm. Our main result disproves a conjecture of Rademacher.

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