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Chen, E, Park, PS and Swaminathan, AA (2016)

On logarithmically Benford Sequences

Proc. Amer. Math. Soc. 144, pp. 4599-4608.

ISSN/ISBN: Not available at this time. DOI: 10.1090/proc/13112



Abstract: Let I ⊂ N be an infinite subset, and let {ai}i∈I be a sequence of nonzero real numbers indexed by I such that there exist positive constants m,C1 for which |ai| ≤ C1 ∑ im for all i ∈ I. Furthermore, let ci ∈ [−1,1] be defined by ci = aim for each i ∈ I, and suppose the ciís are equidistributed C1 ∑i in [−1,1] with respect to a continuous, symmetric probability measure μ. In this paper, we show that if I ⊂ N is not too sparse, then the sequence {ai}i∈I fails to obey Benfordís Law with respect to arithmetic density in any sufficiently large base, and in fact in any base when μ([0,t]) is a strictly convex function of t ∈ (0, 1). Nonetheless, we also provide conditions on the density of I ⊂ N under which the sequence {ai}i∈I satisfies Benfordís Law with respect to logarithmic density in every base. As an application, we apply our general result to study Benfordís 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.


Bibtex:
@article {, AUTHOR = {Evan Chen and Peter S. Park and Ashvin A. Swaminathan}, TITLE = {On logarithmically Benford Sequences}, JOURNAL = {Proc. Amer. Math. Soc.}, YEAR = {2016}, VOLUME = {144}, PAGES = {4599--4608}, DOI = {10.1090/proc/13112 }, URL = {http://www.ams.org/journals/proc/2016-144-11/S0002-9939-2016-13112-2/home.html}, }


Reference Type: Journal Article

Subject Area(s): Number Theory, Probability Theory