# 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.

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