Asymptotics of Reciprocal Supernorm Partition Statistics

2023/08/29 by Lagarias, Jeffrey C., Sun, Chenyang · 1 citation
#05A17 (Secondary) #11P82 (Primary) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2308.15303

Abstract

We consider two multiplicative statistics on the set of integer partitions: the norm of a partition, which is the product of its parts, and the supernorm of a partition, which is the product of the prime numbers p_i indexed by its parts i. We introduce and study new statistics that are sums of reciprocals of supernorms on three statistical ensembles of partitions, labelled by their size |λ|=n, their perimeter equaling n, and their largest part equaling n. We show that the cumulative statistics of the reciprocal supernorm for each of the three ensembles are asymptotic to e^γ log n as n → ∞.

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• Partition-theoretic model of prime distribution, II

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