## On logarithmically Benford Sequences

HTML articles powered by AMS MathViewer

- by Evan Chen, Peter S. Park and Ashvin A. Swaminathan
- Proc. Amer. Math. Soc.
**144**(2016), 4599-4608 - DOI: https://doi.org/10.1090/proc/13112
- Published electronically: July 22, 2016
- PDF | Request permission

## Abstract:

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.

## References

- Theresa C. Anderson, Larry Rolen, and Ruth Stoehr,
*Benford’s law for coefficients of modular forms and partition functions*, Proc. Amer. Math. Soc.**139**(2011), no. 5, 1533–1541. MR**2763743**, DOI 10.1090/S0002-9939-2010-10577-4 - Frank Benford,
*The law of anomalous numbers*, Proc. Amer. Philos. Soc.**78**(1938), no. 4, 551–572. - Arno Berger, Leonid A. Bunimovich, and Theodore P. Hill,
*One-dimensional dynamical systems and Benford’s law*, Trans. Amer. Math. Soc.**357**(2005), no. 1, 197–219. MR**2098092**, DOI 10.1090/S0002-9947-04-03455-5 - Arno Berger and Theodore P. Hill,
*An introduction to Benford’s law*, Princeton University Press, 2015. - E. Hecke,
*Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen*, Math. Z.**6**(1920), no. 1-2, 11–51 (German). MR**1544392**, DOI 10.1007/BF01202991 - Marie Jameson, Jesse Thorner, and Lynnelle Ye,
*Benford’s law for coefficients of newforms*, Int. J. Number Theory**12**(2016), no. 2, 483–494. MR**3461444**, DOI 10.1142/S1793042116500299 - Alex V. Kontorovich and Steven J. Miller,
*Benford’s law, values of $L$-functions and the $3x+1$ problem*, Acta Arith.**120**(2005), no. 3, 269–297. MR**2188844**, DOI 10.4064/aa120-3-4 - Jeffrey C. Lagarias and K. Soundararajan,
*Benford’s law for the $3x+1$ function*, J. London Math. Soc. (2)**74**(2006), no. 2, 289–303. MR**2269630**, DOI 10.1112/S0024610706023131 - Florian Luca, Carl Pomerance, and Štefan Porubský,
*Sets with prescribed arithmetic densities*, Unif. Distrib. Theory**3**(2008), no. 2, 67–80. MR**2480233** - Bruno Massé and Dominique Schneider,
*A survey on weighted densities and their connection with the first digit phenomenon*, Rocky Mountain J. Math.**41**(2011), no. 5, 1395–1415. MR**2838069**, DOI 10.1216/RMJ-2011-41-5-1395 - Steven J. Miller (ed.),
*Benford’s law: theory and applications*, Princeton University Press, 2015. - Ken Ono,
*The web of modularity: arithmetic of the coefficients of modular forms and $q$-series*, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR**2020489** - Kenneth A. Ribet,
*Galois representations attached to eigenforms with Nebentypus*, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., Vol. 601, Springer, Berlin, 1977, pp. 17–51. MR**0453647** - J.-P. Serre,
*A course in arithmetic*, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR**0344216**, DOI 10.1007/978-1-4684-9884-4

## Bibliographic Information

**Evan Chen**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02138
- MR Author ID: 1158569
- Email: evanchen@mit.edu
**Peter S. Park**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 1046046
- Email: pspark@math.princeton.edu
**Ashvin A. Swaminathan**- Affiliation: Department of Mathematics, Harvard College, Cambridge, Massachusetts 02138
- Email: aaswaminathan@college.harvard.edu
- 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: https://doi.org/10.1090/proc/13112
- MathSciNet review: 3544512