The Dirichlet L-function is a complex function that generalizes the Riemann zeta function and plays a crucial role in number theory, particularly in the study of Dirichlet characters and L-series. It is associated with a Dirichlet character \( \chi \) modulo \( k \), which is a completely multiplicative arithmetic function satisfying certain periodicity and the condition \( \chi(n) = 0 \) for \( n \) not coprime to \( k \).
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For a Dirichlet character modulo , its Dirichlet L-function is initially defined on the half-plane bySince , this series converges absolutely by comparison with .
A Dirichlet character is a completely multiplicative function. The fundamental theorem of arithmetic therefore reorganizes the absolutely convergent series into an Euler product:Expanding each factor as a geometric series assigns exactly one term to every positive integer. For the principal Dirichlet character , the factors at primes dividing disappear, giving , where is the Riemann zeta function.