Prime Numbers, Dirichlet Density, and Benford's Law

TL;DR

This paper investigates the relationship between prime numbers and Benford's Law, revealing that primes do not truly follow Benford's Law despite certain density results, and explores their digital behavior and logarithmic density.

Contribution

It critically analyzes the Dirichlet density for primes and demonstrates that primes cannot be considered Benford, providing new insights into their digital distribution and density behavior.

Findings

Prime numbers do not obey Benford's Law in the usual sense.

Dirichlet density results are superficial and do not imply Benford conformity.

Logarithmic density of primes increases and is confirmed empirically.

Abstract

The Prime Numbers are well-known for their paradoxical stand regarding Benford's Law. On one hand they adamantly refuse to obey the law of Benford in the usual sense, namely that of a normal density of the proportion of primes with d as the leading digit, yet on the other hand, the Dirichlet density for the subset of all primes with d as the leading digit is indeed LOG(1 + 1/d). In this article the superficiality of the Dirichlet density result is demonstrated and explained in terms of other well-known and established results in the discipline of Benford's Law, conceptually concluding that prime numbers cannot be considered Benford at all, in spite of the Dirichlet density result. In addition, a detailed examination of the digital behavior of prime numbers is outlined, showing a distinct digital development pattern, from a slight preference for low digits at the start for small primes, to a complete digital equality for large primes in the limit as the prime number sequence goes to infinity. Finally an exact analytical expression for the density of the logarithms of primes is derived and shown to be always on the rise at the macro level, an observation that is also confirmed empirically.