High and odd moments in the Erdős--Kac theorem

2024/12/31 by Ofir Gorodetsky, Gorodetsky, Ofir
Mathematics · Computer Science · #Analytic Number Theory Research #Algorithms and Data Compression

paper · pdf · doi:10.48550/arxiv.2501.00351

Abstract

Granville and Soundararajan showed that the kth moment in the Erdős--Kac theorem is equal to the kth moment of the standard Gaussian distribution in the range k=o((log log x)^1/3), up to a negligible error term. We show that their range is sharp: when k/(log log x)^1/3 tends to infinity, a different behavior emerges, and odd moments start exhibiting similar growth to even moments. For odd k we find the asymptotics of the kth moment when k=O((log log x)^1/3), where previously only an upper bound was known. Our methods are flexible and apply to other distributions, including the Poisson distribution, whose centered moments turn out to be excellent approximations for the Erdős--Kac moments.

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