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arXiv · 1803.02178

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

Michael Drmota·Christian Krattenthaler
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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.

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Michael Drmota, Christian Krattenthaler. 2018-03-06. A Joint Central Limit Theorem for the Sum-of-Digits Function, and Asymptotic Divisibility of Catalan-like Sequences. https://arxiv.org/abs/1803.02178

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