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Pollack, P and Roy, AS (2022)

Dirichlet, Sierpiński, and Benford

Journal of Number Theory (pre-proof).

ISSN/ISBN: Not available at this time. DOI: 10.1016/j.jnt.2021.12.010

Abstract: Sixty years ago, Sierpiński observed that for any positive integers A and B, and any g ≥ 2, there are infinitely many primes whose base g-expansion begins with the digits of A and ends with those of B. Sierpiński’s short proof rests on the prime number theorem for arithmetic progressions (PNT for APs). We explain how his result can be viewed as a natural intermediary between Dirichlet’s theorem on primes in progressions and the PNT for APs. In addition to being of pedagogical interest, this perspective quickly yields a generalization of Sierpiński’s result where the initial and terminal digits of p are prescribed in two coprime bases simultaneously; moreover, the proportion (Dirichlet density) of the corresponding primes is determined explicitly. The same quasielementary method shows that the arithmetic functions φ(n), σ(n), and d(n) obey “Benford’s law” in a suitable sense.

@artcile{, author = {Paul Pollack and Akash Singha Roy}, title = {Dirichlet, Sierpinski, and Benford}, year = {2022}, journal = {Journal of Number Theory}, doi = {10.1016/j.jnt.2021.12.010}, url = {}, }

Reference Type: Journal Article

Subject Area(s): Number Theory