A new proof of the Erdos-Kac central limit theorem

• Published in: Transactions of the American Mathematical Society 2024-02, Vol.377 (2), p.1475-1503
• Main Authors: Cranston, Michael, Mountford, Thomas
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/9075

In this paper we use the Riemann zeta distribution to give a new proof of the Erdös-Kac Central Limit Theorem. That is, if ζ ( s ) = ∑ n ≥ 1 1 n s \zeta (s)=\sum _{n\ge 1} \frac {1}{n^s} , s > 1 , s>1, then we consider the random variable X s X_s with P ( X s = n ) = 1 ζ ( s ) n s , P(X_s=n)=\frac {1}{\zeta (s)n^s}, n ≥ 1. n\ge 1. In an earlier paper, the first author and Adrien Peltzer derived the analog of the Erdös-Kac Central Limit Theorem (CLT) for the number of distinct prime factors, ω ( X s ) , \omega (X_s), of X s , X_s, as s ↘ 1. s\searrow 1. In this paper we show, by means of a Tauberian Theorem, how to obtain the Central Limit Theorem of Erdös-Kac for the uniform distribution from the result for the random variable X s X_s . We also apply the technique to the number of distinct prime divisors of X s X_s that lie in an arithmetic sequence and a local CLT of the type proved by Dixit and Murty [Hardy-Ramanujan J. 43 (2020), 17–23] as well a version of the CLT for irreducible divisors of a monic polynomial over a finite field.