### Cai, Z, Faust, M, Hildebrand, AJ, Li, J and Zhang, Y (2019)

#### The Surprising Accuracy of Benford’s Law in Mathematics

Preprint arXiv:1907.08894 [math.PR]; last accessed July 31, 2019.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

**Abstract:** Benford’s law is an empirical “law” governing the frequency of leading dig- its in numerical data sets. Surprisingly, for mathematical sequences the predictions de- rived 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 109 log10 2 = 301029995.66 . . . . 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.

**Bibtex:**

```
@ARTICLE{,
author = {{Cai}, Zhaodong and {Faust}, Matthew and {Hildebrand}, A.~J. and
{Li}, Junxian and {Zhang}, Yuan},
title = "{The Surprising Accuracy of Benford's Law in Mathematics}",
journal = {arXiv e-prints},
year = "2019",
month = "Jul",
eprint = {1907.08894},
primaryClass = {math.PR},
}
```

**Reference Type:** Preprint

**Subject Area(s):** Number Theory