High and odd moments in the Erd\H{o}s--Kac theorem

TL;DR

This paper investigates the behavior of higher moments in the Erdős–Kac theorem, revealing a phase transition in odd moments beyond a certain range and providing precise asymptotics for these moments.

Contribution

It establishes the sharp boundary for the Gaussian approximation of moments and derives asymptotics for odd moments, extending previous bounds and applying methods to other distributions.

Findings

Odd moments exhibit similar growth to even moments beyond the known range.

Asymptotics for the $k$th moment when $k=O(( ext{log log } x)^{1/3})$ are derived.

Centered Poisson moments approximate Erdős–Kac moments well.

Abstract

Granville and Soundararajan showed that the $k$th moment in the Erd\H{o}s--Kac theorem is equal to the $k$th 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 $k$th 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 Erd\H{o}s--Kac moments.