Dirichlet L-function

A Dirichlet L-function is a meromorphic function on the complex plane attached to a Dirichlet character χ modulo q, defined by the series L(s, χ) = Σ_{n=1}^∞ χ(n) / n^s for Re(s) > 1. When χ is the principal character, L(s, χ) reduces to the Riemann zeta function multiplied by a simple Euler factor; for non-principal characters, L(s, χ) is entire. These functions were introduced by Johann Peter Gustav Lejeune Dirichlet to prove his celebrated theorem on primes in arithmetic progressions, and they remain the simplest examples of Artin L-functions and Hecke L-functions.

The Dirichlet L-function is where analytic number theory began. Before Dirichlet, number theory was a discipline of congruences and Diophantine equations; after Dirichlet, it was a discipline of complex analysis. The L-function is not a tool applied to number theory — it is the moment number theory became analysis.