[Paper Review] Sieving and the Erd{\H o}s-Kac theorem

Andrew Granville, K. Soundararajan|ArXiv.org|Jun 1, 2006

Analytic Number Theory Research20 references3 citations

TL;DR

This paper provides a streamlined proof of the Erdős-Kac theorem using moment computations, establishing that the distribution of the number of distinct prime factors ω(n) is asymptotically normal with mean log log n and variance log log n. The authors extend this method to general sieve-theoretic settings, deriving precise asymptotic formulas for moments of ω(n) and showing they match those of a normal distribution, thereby offering a unified framework for Erdős-Kac-type theorems across additive arithmetic functions.

ABSTRACT

We give a relatively easy proof of the Erd\H os-Kac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature.

Motivation & Objective

• To provide a simplified, moment-based proof of the Erdős-Kac theorem, avoiding reliance on complex analytic or probabilistic methods.
• To extend the moment method to general arithmetic sequences under sieve-theoretic hypotheses, generalizing the Erdős-Kac result beyond the classical case.
• To derive precise asymptotic formulas for the k-th moments of ω(n) − log log x, showing they match those of a normal distribution.
• To unify and generalize existing results on the distribution of additive arithmetic functions, including those of Kubilius and Shapiro.
• To demonstrate that the moment method is robust in sieve-theoretic contexts, enabling new applications to number-theoretic distribution problems.

Proposed method

• The authors define a function fp(n) that encodes divisibility by prime p, enabling the construction of a sum over p ≤ z of fp(n), which models the fluctuation of ω(n) around its mean.
• They compute the k-th moment of ∑p≤z fp(n) using combinatorial expansion and sieve-type error estimates, leveraging the independence structure of primes.
• Key estimates involve the variance σP(g)² and the mean μP(g), with error terms controlled via the size of g(p) and the sum of h(p)/p over primes.
• The main term arises from pairings of primes (k even), corresponding to the Gaussian moment structure, while higher-order terms are bounded using factorial and multinomial coefficients.
• The method uses a technical proposition (Proposition 2) that gives asymptotic formulas for ∑n≤x(∑p≤z fp(n))k with explicit error terms depending on k and log log z.
• The framework is generalized to arbitrary additive functions via a weighted sum ∑p∈P g(p)fp(n), allowing application to non-constant or irregular arithmetic functions.

Experimental results

Research questions

• RQ1 Can the Erdős-Kac theorem be proven via moment computations rather than probabilistic or analytic methods?
• RQ2 How do the k-th moments of ω(n) − log log x behave asymptotically, and do they match those of a normal distribution?
• RQ3 To what extent can the moment method be extended to general additive arithmetic functions under sieve-theoretic assumptions?
• RQ4 What conditions ensure that the distribution of an additive function converges to a normal law, and how can this be quantified?
• RQ5 Can the results of Kubilius and Shapiro on the distribution of additive functions be recovered and generalized using this moment-based sieve framework?

Key findings

• For even k ≤ (log log x)^{1/3}, the k-th moment ∑n≤x(ω(n) − log log x)^k is asymptotically C_k x (log log x)^{k/2} (1 + O(k^{3/2}/√(log log x))), where C_k = Γ(k+1)/(2^{k/2} Γ(k/2 + 1)).
• For odd k ≤ (log log x)^{1/3}, the k-th moment is bounded by C_k x (log log x)^{k/2} times O(k^{3/2}/√(log log x)).
• The main term in the moment expansion arises from pairings of distinct primes, corresponding to the Gaussian moment structure, and dominates when k is even.
• The error terms are controlled by O(2^k π(z)^k) and depend on the size of g(p) and the sum of h(p)/p, ensuring uniformity in k.
• The method generalizes to arbitrary additive functions g(n) = ∑p|n g(p), with the distribution of ∑p∈P g(p)fp(n) converging to normality under suitable sieve conditions.
• The framework recovers and extends results of Kubilius and Shapiro, showing that the Erdős-Kac theorem holds for a broad class of additive functions under mild arithmetic conditions.