# The Surprising Accuracy of Benford's Law in Mathematics

## TL;DR

Benford's law accurately predicts leading digit distributions in mathematical sequences, with the paper providing proofs and explanations for these surprising accuracies using number theory and Diophantine approximation.

## Contribution

The paper demonstrates that Benford's law applies to mathematical sequences and offers new proofs and explanations for the observed accuracy, connecting it to classical and modern number theory.

## Key findings

- Benford's law accurately predicts digit frequencies in powers of integers.
- Mathematical sequences exhibit near-perfect adherence to Benford's law.
- The work links digit distribution phenomena to Diophantine approximation and conjectures.

## 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 $10^9\log_{10}2=301029995.66\dots$. 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.