 Dirichlet L-function

For a Dirichlet character $\chi$ modulo $q$, its Dirichlet L-function is initially defined on the half-plane $\mathrm{Re}s>1$ bySince $|\chi (n)|\le 1$, this series converges absolutely by comparison with ${\sum }_n{n}^{-\mathrm{Re}s}$.

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 ${\chi }_0$, the factors at primes dividing $q$ disappear, giving $L(s,{\chi }_0)=\zeta (s){\prod }_{p\mid q}(1-p^{-s})$, where $\zeta$ is the Riemann zeta function.

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 \( k \).