The Surprising Accuracy of Benford’s Law in Mathematics

Abstract

Benford’s law is an empirical “law” governing the frequency of leading digits in numerical data sets. Surprisingly, for mathematical sequences the predictions derived from it can be uncannily accurate. For example, among the first billion powers of 2, exactly 301029995 begin with digit 1, while the Benford prediction for this count is ${10}^9{\hspace{0.17em}\text{log}\hspace{0.17em}}_{10}2=301029995.66\dots$. Similar “perfect hits” can be observed in other instances, such as the digit 1 and 2 counts for the first billion powers of 3. We prove results that explain many, but not all, of these surprising accuracies, and we relate the observed behavior to classical results in Diophantine approximation as well as recent deep conjectures in this area.

• MSC: Primary 60F99
• Secondary 11K31
• 11Y55

Acknowledgments

We are grateful to the referees for their careful reading of the paper and helpful suggestions and comments. This work originated with an undergraduate research project carried out in 2016 at the Illinois Geometry Lab (IGL) at the University of Illinois; we thank the IGL for providing this opportunity.

Additional information

Notes on contributors

Zhaodong Cai

ZHAODONG CAI received his B.S. in mathematics from the University of Illinois in 2017 and is currently a Ph.D. student at the University of Pennsylvania. When not studying mathematics, he likes solving chess puzzles.

Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab, 209 South 33rd St., Philadelphia, PA 19104, USA [email protected]

Matthew Faust

MATTHEW FAUST is a Ph.D. student studying mathematics at Texas A&M University. He received B.S. degrees in computer engineering and mathematics in 2018 from the University of Illinois. He is interested in algebraic combinatorics and algebraic geometry. In his free time, he enjoys strategy games, preferably cooperative with Yuan Zhang.

Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843, USA [email protected]

A. J. Hildebrand

A. J. HILDEBRAND received his Ph.D. in mathematics from the University of Freiburg in 1983 and has been at the University of Illinois since 1986, becoming professor emeritus in 2012. Since retiring from the University of Illinois, he has supervised over 100 undergraduates on research projects in pure and applied mathematics and allied areas. The present article grew out of one of these projects.

Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801, USA [email protected]

Junxian Li

JUNXIAN LI received her Ph.D. in 2018 from the University of Illinois under the supervision of Alexandru Zaharescu. Her research interests are in number theory. During her Ph.D. studies, she has enjoyed doing research with enthusiastic undergraduates as a graduate mentor in the Illinois Geometry Lab. She is currently a postdoc at the Max Planck Institute for Mathematics.

Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany [email protected]

Yuan Zhang

YUAN ZHANG began his college education at the University of Illinois as a major in natural resources and environment sciences. He soon changed his major to mathematics, receiving his B.S. degree in 2018. He is currently a Ph.D. student studying mathematics at the University of Virginia. He is interested in algebraic topology and algebraic combinatorics. In his free time, he enjoys playing strategy games, especially with his friend Matt Faust.

Department of Mathematics, University of Virginia, 141 Cabell Dr., Kerchof Hall, Charlottesville, VA 22904, USA [email protected]