The concept of "character" in Dirichlet's theorem on primes in an arithmetic progression

TL;DR

This paper surveys the historical development and use of Dirichlet characters in the proof of Dirichlet's theorem on primes in arithmetic progressions, highlighting how mathematical methods evolved over time.

Contribution

It provides a historical analysis of the role of Dirichlet characters, revealing the pragmatic influences on the development of modern number theory methods.

Findings

Dirichlet characters were central to early proofs of the theorem

Historical use of characters influenced the evolution of proof techniques

The survey clarifies the pragmatic pressures shaping mathematical methods

Abstract

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses of Dirichlet characters in presentations of Dirichlet's proof in the nineteenth and early twentieth centuries, with an eye towards understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method.