High and Odd Moment Asymptotics in the Erdos-Kac Theorem
Beyond the classical Erdos-Kac Gaussian limit for the prime-factor-counting function omega(n), this determines the asymptotics of its high centered moments, including odd-order moments, throughout the range k = O(log log x). A saddle-point modification of the Sathe-Selberg method extracts the moments from their generating function, showing the Poisson distribution of mean log log x supplies the correct approximation and that the Gaussian range k = o((log log x)^{1/3}) is sharp.
2501.00351
The Erdos-Kac theorem states that the number of distinct prime factors omega(n) of a random integer n <= x, once centered by log log x and scaled by sqrt(log log x), converges in distribution to a st…