Journal of Number Theory (pre-proof).

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

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**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.

**Bibtex:**

```
@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 = {https://www.sciencedirect.com/science/article/pii/S0022314X22000099},
}
```

**Reference Type:** Journal Article

**Subject Area(s):** Number Theory