A joint central limit theorem for the sum-of-digits function, and asymptotic divisibility of Catalan-like sequences

Abstract:

We prove a central limit theorem for the joint distribution of $s_q(A_jn)$, $1\le j \le d$, where $s_q$ denotes the sum-of-digits function in base $q$ and the $A_j$’s are positive integers relatively prime to $q$. We do this in fact within the framework of quasi-additive functions. As an application, we show that most elements of “Catalan-like” sequences—by which we mean integer sequences defined by products/quotients of factorials—are divisible by any given positive integer.