### Cai, Z, Faust, M, Hildebrand, AJ, Li, J and Zhang, Y (2020)

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

The American Mathematical Monthly 127(3), pp. 217-237.

**ISSN/ISBN:** Not available at this time.
**DOI:** 10.1080/00029890.2020.1690387

**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 log_10 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 = {Zhaodong Cai and Matthew Faust and A. J. Hildebrand and Junxian Li and Yuan Zhang},
title = {The Surprising Accuracy of Benford's Law in Mathematics},
journal = {The American Mathematical Monthly},
volume = {127},
number = {3},
pages = {217-237},
year = {2020},
publisher = {Taylor & Francis},
doi = {10.1080/00029890.2020.1690387},
url = {https://www.tandfonline.com/doi/full/10.1080/00029890.2020.1690387},
}
```

**Reference Type:** Journal Article

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