An Erd\H{o}s-Kac theorem for integers with dense divisors

TL;DR

This paper proves that integers with dense divisors have their number of prime factors distributed approximately normally, extending the Erdős-Kac theorem to a new class of integers with bounded divisor ratios.

Contribution

It establishes a normal distribution law for prime factors of integers with dense divisors and generalizes the classical Erdős-Kac theorem to this setting.

Findings

Number of prime factors follows a normal distribution for dense divisor integers

Mean of distribution is approximately 2.280 times log log n

Variance of distribution is approximately 0.414 times log log n

Abstract

We show that for large integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C \log_2 n$ and variance $V \log_2 n$, where $C=1/(1-e^{-\gamma})\approx 2.280$ and $V\approx 0.414$. This result is then generalized in two different directions.