Transforming the Erd\H{o}s-Kac theorem

TL;DR

This paper introduces new transformations and interval estimation methods for the prime omega function, improving accuracy and reliability, especially for small integers, by applying the delta method and analyzing coverage probabilities.

Contribution

It proposes novel transformations and interval estimates for the prime omega function, including the score and Poisson intervals, with theoretical and empirical validation.

Findings

Square-root and three-quarters power transformations stabilize variance.

Score interval estimate performs well for small positive integers.

Poisson interval estimate is reliably effective regardless of training.

Abstract

Transforming the Erd\H{o}s-Kac theorem provides more flexibility in how the theorem can be utilized as an interval estimate for the prime omega function, which counts the number of distinct prime divisors. Here, we consider a direct transformation by the delta method. Then, we demonstrate that the square-root and three-quarters power asymptotically achieve variance stabilization and an optimal width, respectively. Furthermore, by adjusting the denominator of the theorem, we derive the score interval estimate. To make these interval estimates reliable for small positive integers, we examine performances of various interval estimates for the prime omega function using fuzzy coverage probabilities. The results indicate that the score interval estimate performs well even for small positive integers after training the mean and standard deviation using the prime omega function. Moreover, the Poisson interval estimate is relatively reliable with or without training. Additional theoretical results on the Erd\H{o}s-Pomerance theorem are also provided.