The Surprising Accuracy of Benford’s Law in Mathematics

@article{Cai2020TheSA,
  title={The Surprising Accuracy of Benford’s Law in Mathematics},
  author={Zhaodong Cai and Matthew Faust and A. J. Hildebrand and Junxian Li and Yali Zhang},
  journal={The American Mathematical Monthly},
  year={2020},
  volume={127},
  pages={217 - 237}
}
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 . 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… 

References

SHOWING 1-10 OF 29 REFERENCES
Leading Digits of Mersenne Numbers
TLDR
This article presents data, based on the first billion terms of the sequence , showing that leading digits of Mersenne numbers behave in many respects more regularly than those in the above smooth sequences.
Benford's Law, A Growth Industry
  • K. Ross
  • Economics, Mathematics
    Am. Math. Mon.
  • 2011
TLDR
This paper provides a simple explanation, suitable for nonmathematicians, of why Benford's law holds for data that have been growing (or shrinking) exponentially over time.
Randomness of the square root of 2 and the giant leap, part 2
  • J. Beck
  • Mathematics
    Period. Math. Hung.
  • 2011
TLDR
It is proved that the “quadratic irrational rotation” exhibits a central limit theorem, and the distribution of this renormalized counting number is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as N tends to infinity.
The strong law of small numbers
This article is in two parts, the first of which is a do-it-yourself operation, in which I'll show you 35 examples of patterns that seem to appear when we look at several small values of n, in
Randomness of the square root of 2 and the Giant Leap, Part 1
  • J. Beck
  • Mathematics
    Period. Math. Hung.
  • 2010
TLDR
The proof is rather complicated and long; it has many interesting detours and byproducts; the exact determination of the key constant factors (in the additive and multiplicative norming), which depend on α and x, requires surprisingly deep algebraic tools.
Uniform distribution of sequences
( 1 ) {xn}z= Xn--Z_I Zin-Ztn-I is uniformly distributed mod 1, i.e., if ( 2 ) lim (1/N)A(x, N, {xn}z)-x (0x<_ 1), where A(x, N, {Xn)) denotes the number of indices n, l<=n<=N such that {x} is less
The Distribution of Leading Digits and Uniform Distribution Mod 1
Randomness in lattice point problems
  • J. Beck
  • Mathematics
    Discret. Math.
  • 2001
...
...