Modeling the Dirichlet distribution using multiplicative functions

Abstract

For q,m,n,d ∈ N and some multiplicative function f > 0, we denote by T3(n) the sum of f(d) over the ordered triples (q,m,d) with qmd = n. We prove that Cesaro mean of distribution functions defined by means of T3 uniformly converges to the one-parameter Dirichlet distribution function. The parameter of the limit distribution depends on the values of f on primes. The remainder term is estimated as well.