### abstract

- Consider the multiplicities e p 1 ( n ) , e p 2 ( n ) , … , e p k ( n ) in which the primes p 1 , p 2 , … , p k appear in the factorization of n !. We show that these multiplicities are jointly uniformly distributed modulo ( m 1 , m 2 , … , m k ) for any fixed integers m 1 , m 2 , … , m k , thus improving a result of Luca and Stănică [F. Luca, P. Stănică, On the prime power factorization of n !, J. Number Theory 102 (2003) 298–305]. To prove the theorem, we obtain a result regarding the joint distribution of several completely q -additive functions, which seems to be of independent interest.