Daniel Berend

Consider the multiplicities e_p1 (n), e_p2 (n), ..., e_pk (n) in which the primes p₁, p₂, ..., p_k appear in the factorization of n!. We show that these multiplicities are jointly uniformly distributed modulo (m₁, m₂, ..., m_k) for any fixed integers m₁, m₂, ..., m_k, thus improving a result of Luca and Stǎnicǎ [F. Luca, P. Stǎnicǎ, 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.