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Luque, B and Lacasa, L (2009)

The first-digit frequencies of prime numbers and Riemann zeta zeros

Proc. Royal Soc. A, published online 22Apr09.

ISSN/ISBN: Not available at this time. DOI: 10.1098/rspa.2009.0126

Abstract: Prime numbers seem to be distributed among the natural numbers with no law other than that of chance; however, their global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated scientists across the ages to search for local and global patterns in this distribution that could eventually shed light on the ultimate nature of primes. In this paper, 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 datasets, describes with astonishing precision the statistical distribution of leading digits in the prime number sequence. Moreover, a reciprocal version of this pattern also takes place in the sequence of the non-trivial Riemann zeta zeros. We prove that the prime number theorem is, in the final analysis, responsible for these patterns.

@inproceedings{, title={The first-digit frequencies of prime numbers and Riemann zeta zeros}, author={Luque, Bartolo and Lacasa, Lucas}, booktitle={Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences}, pages={rspa--2009}, year={2009}, organization={The Royal Society}, DOI={10.1098/rspa.2009.0126}, }

Reference Type: Journal Article

Subject Area(s): Number Theory