Analysis and Applications, pp. 1-24.
ISSN/ISBN: Not available at this time. DOI: 10.1142/S0219530524500398
Abstract: This paper develops a new description of the asymptotics for the empirical distributions of significands and significant digits associated with (pn), where pn denotes the nth prime number. The work utilizes the space of probability measures on the significand, endowed with a suitable Kantorovich metric, as well as finite-dimensional projections thereof. For sequences sufficiently close to (pn), it is shown that the limit points of the associated empirical distributions form a circle that is made up of all rescalings of a single absolutely continuous distribution, and is centered at a distribution known as Benford’s law (BL). The precise rate of convergence to that circle is determined. Moreover, even in the infinite-dimensional setting of significands the convergence is seen to occur along a distinguished low-dimensional object, in fact, along a smooth curve intimately related to BL. By connecting (pn) and BL in a new way, the results rigorously confirm well-documented experimental observations and complement known facts in the literature.
Bibtex:
@article{,
author = {Berger, Arno and Rahmatidehkordi, Ardalan},
title = {The primes perform a Benford dance},
journal = {Analysis and Applications},
volume = {0},
number = {0},
pages = {1-24},
year = {2024},
doi = {10.1142/S0219530524500398},
URL = {https://doi.org/10.1142/S0219530524500398},
}
Reference Type: Journal Article
Subject Area(s): Number Theory