Dirichlet, Sierpiński, and Benford

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.