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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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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On logarithmically Benford Sequences
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by Evan Chen, Peter S. Park and Ashvin A. Swaminathan PDF
Proc. Amer. Math. Soc. 144 (2016), 4599-4608 Request permission


Let $\mathcal {I} \subset \mathbb {N}$ be an infinite subset, and let $\{a_i\}_{i \in \mathcal {I}}$ be a sequence of nonzero real numbers indexed by $\mathcal {I}$ such that there exist positive constants $m, C_1$ for which $|a_i| \leq C_1 \cdot i^m$ for all $i \in \mathcal {I}$. Furthermore, let $c_i \in [-1,1]$ be defined by $c_i = \frac {a_i}{C_1 \cdot i^m}$ for each $i \in \mathcal {I}$, and suppose the $c_i$’s are equidistributed in $[-1,1]$ with respect to a continuous, symmetric probability measure $\mu$. In this paper, we show that if $\mathcal {I} \subset \mathbb {N}$ is not too sparse, then the sequence $\{a_i\}_{i \in \mathcal {I}}$ fails to obey Benford’s Law with respect to arithmetic density in any sufficiently large base, and in fact in any base when $\mu ([0,t])$ is a strictly convex function of $t \in (0,1)$. Nonetheless, we also provide conditions on the density of $\mathcal {I} \subset \mathbb {N}$ under which the sequence $\{a_i\}_{i \in \mathcal {I}}$ satisfies Benford’s Law with respect to logarithmic density in every base.

As an application, we apply our general result to study Benford’s Law-type behavior in the leading digits of Frobenius traces of newforms of positive, even weight. Our methods of proof build on the work of Jameson, Thorner, and Ye, who studied the particular case of newforms without complex multiplication.

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Additional Information
  • Evan Chen
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02138
  • MR Author ID: 1158569
  • Email:
  • Peter S. Park
  • Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
  • MR Author ID: 1046046
  • Email:
  • Ashvin A. Swaminathan
  • Affiliation: Department of Mathematics, Harvard College, Cambridge, Massachusetts 02138
  • Email:
  • Received by editor(s): January 8, 2016
  • Published electronically: July 22, 2016
  • Communicated by: Ken Ono
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 4599-4608
  • MSC (2010): Primary 11F11, 11N05
  • DOI:
  • MathSciNet review: 3544512