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High and odd moments in the Erdős--Kac theorem

Ofir Gorodetsky

TL;DR

The paper investigates high and odd moments in the Erdős–Kac setting, clarifying when the centered count $\frac{\omega(n)-\log\log x}{\sqrt{\log\log x}}$ behaves like a Gaussian and when Poisson-type deviations take over. It develops a saddle-point–generating-function framework to compare Erdős–Kac moments with those of Poisson and Bernoulli sums, yielding precise asymptotics for $k$ up to $O(\log\log x)$ and identifying a sharp transition at $k\sim (\log\log x)^{1/3}$. The results include new asymptotics for odd moments in the regime $k=O((\log\log x)^{1/3})$ and demonstrate that Meissel–Mertens constants and Poisson-Bernoulli models provide excellent approximations for the Erdős–Kac moments. The methods are robust and extend to other distributions, highlighting the broader applicability of generating-function and saddle-point techniques in probabilistic number theory.

Abstract

Granville and Soundararajan showed that the $k$th moment in the Erdős--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 Erdős--Kac moments.

High and odd moments in the Erdős--Kac theorem

TL;DR

The paper investigates high and odd moments in the Erdős–Kac setting, clarifying when the centered count behaves like a Gaussian and when Poisson-type deviations take over. It develops a saddle-point–generating-function framework to compare Erdős–Kac moments with those of Poisson and Bernoulli sums, yielding precise asymptotics for up to and identifying a sharp transition at . The results include new asymptotics for odd moments in the regime and demonstrate that Meissel–Mertens constants and Poisson-Bernoulli models provide excellent approximations for the Erdős–Kac moments. The methods are robust and extend to other distributions, highlighting the broader applicability of generating-function and saddle-point techniques in probabilistic number theory.

Abstract

Granville and Soundararajan showed that the th moment in the Erdős--Kac theorem is equal to the th moment of the standard Gaussian distribution in the range , up to a negligible error term. We show that their range is sharp: when tends to infinity, a different behavior emerges, and odd moments start exhibiting similar growth to even moments. For odd we find the asymptotics of the th moment when , 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.
Paper Structure (19 sections, 17 theorems, 97 equations)

This paper contains 19 sections, 17 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Remark on primes in short intervals
  4. Full results
  5. Proofs of theorems
  6. Proof of Theorem \ref{['thm:main']}
  7. Proof of Theorem \ref{['thm:lowkpo']}
  8. Proof of Theorem \ref{['thm:lowk']}
  9. Proof of Theorem \ref{['thm:lowkpobern']}
  10. Generating function lemmas
  11. Proof of Propositions \ref{['prop:babypoisson']}--\ref{['prop:comp']}
  12. Proof of Propositions \ref{['prop:omega']}--\ref{['prop:kac2']}
  13. Proof of Proposition \ref{['prop:pb']}
  14. Proofs of Generating function lemmas
  15. Proof of Lemma \ref{['lem:tech0']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Fix $A>0$. Suppose $x\ge 3$ and $2 \le k \le A\log \log x$. Define $r>0$ via If $k=o(\log \log x)$ then $r=o(1)$ and so $F(e^r)/e^{ra}\sim e^{-\gamma (e^r-1)}/\Gamma(e^r) \sim 1$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1
  • Remark
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 10 more