The first digit frequencies of primes and Riemann zeta zeros tend to uniformity following a size-dependent generalized Benford's law

TL;DR

This paper demonstrates that the distribution of leading digits in primes and Riemann zeta zeros follows a size-dependent generalized Benford's law, revealing underlying regularities linked to prime distribution and the prime number theorem.

Contribution

It introduces a generalized Benford's law that accurately models the leading digit distribution of primes and zeta zeros, connecting these patterns to prime number theory.

Findings

Prime leading digits follow a size-dependent generalized Benford's law.

Riemann zeta zeros exhibit a reciprocal pattern consistent with this law.

New approximation to the prime counting function pi(n) is derived.

Abstract

Prime numbers seem to distribute among the natural numbers with no other law than that of chance, however its global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated sci- entists of all ages to search for local and global patterns in this distribution that eventually could shed light into the ultimate nature of primes. In this work we show that a generalization of the well known first-digit Benford's law, which addresses the rate of appearance of a given leading digit d in data sets, describes with astonishing precision the statistical distribution of leading digits in the prime numbers sequence. Moreover, a reciprocal version of this pattern also takes place in the sequence of the nontrivial Riemann zeta zeros. We prove that the prime number theorem is, in the last analysis, the responsible of these patterns. Some new relations concerning the prime numbers distribution are also deduced, including a new approximation to the counting function pi(n). Furthermore, some relations concerning the statistical conformance to this generalized Benford's law are derived. Some applications are finally discussed.