Regularity of patterns in the factorization of n!

Consider the multiplicities ep1(n),ep2(n),…,epk(n) in which the primes p1,p2,…,pk appear in the factorization of n!. We show that these multiplicities are jointly uniformly distributed modulo (m1,m2,…,mk) for any fixed integers m1,m2,…,mk, 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.