A Joint Central Limit Theorem for the Sum-of-Digits Function, and Asymptotic Divisibility of Catalan-like Sequences

TL;DR

This paper establishes a joint central limit theorem for sum-of-digits functions in base q and demonstrates that most elements of Catalan-like sequences are divisible by any positive integer, revealing new probabilistic and divisibility properties.

Contribution

It introduces a joint CLT for sum-of-digits functions within quasi-additive functions and applies it to prove divisibility properties of Catalan-like sequences.

Findings

Joint CLT for sum-of-digits functions in multiple bases

Most Catalan-like sequence elements are divisible by any integer

Framework of quasi-additive functions for digit sum analysis

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 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.