# Journal of the Ramanujan Mathematical Society

Volume 37, Issue 2, June 2022  pp. 147–157.

A central limit theorem associated with square-full divisor function in short intervals

Authors:  Xin Li and Kui Liu
Author institution:Department of Mathematics, China University of Mining and Technology, Beijing 10083, People's Republic of China

Summary:  Let g(n) be the number of square-full divisors of integer n. In this paper, we prove a Central Limit Theorem (Erdös-Kac type) weighted by g(n) in short intervals and show that the error term appearing in the asymptotic formula in the theorem is optimal. We also prove an asymptotic formula for the mean value of g(n) in short intervals with n satisfying ω(n) = m, where ω(n) is the number of distinct prime divisors of n.