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Pollack, P and Roy, AS (2021)

Dirichlet, Sierpinski, and Benford

Unpublished manuscript.

ISSN/ISBN: Not available at this time. DOI: Not available at this time.

Abstract: Sixty years ago, Sierpin ́ski 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. Sierpin ́ski’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 Sierpin ́ski’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.

@misc{, author = {Paul Pollack and Akash Singha Roy}, title = {Dirichlet, Sierpinski, and Benford}, year = {2021}, url = {}, }

Reference Type: Other

Subject Area(s): Number Theory