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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A joint central limit theorem for the sum-of-digits function, and asymptotic divisibility of Catalan-like sequences
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by Michael Drmota and Christian Krattenthaler PDF
Proc. Amer. Math. Soc. 147 (2019), 4123-4133 Request permission

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.
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Additional Information
  • Michael Drmota
  • Affiliation: Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria
  • MR Author ID: 59890
  • Christian Krattenthaler
  • Affiliation: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
  • MR Author ID: 106265
  • Received by editor(s): March 30, 2018
  • Received by editor(s) in revised form: August 8, 2018
  • Published electronically: July 8, 2019
  • Additional Notes: This research was partially supported by the Austrian Science Foundation FWF, in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”, project F50-N15.
  • Communicated by: Patricia L. Hersh
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4123-4133
  • MSC (2010): Primary 11K65; Secondary 05A15, 11A63, 11N60, 60F05
  • DOI: https://doi.org/10.1090/proc/14349
  • MathSciNet review: 4002530