In 1837, Lejeune Dirichlet generalized Euler's zeta function (defined in terms of Euler zeta series) into Dirichlet L-functions (defined in terms of Dirichlet L-series) to prove that in any arithmetic progression , where and are coprime, there are infinitely many primes (i.e. there are infinitely many primes for each residue class coprime to )
where is the Dirichlet character of .
The Dirichlet character is a function of , i.e. it is a function of the remainder (residue class) of when divided by and it must be 0 when and are not coprime. It must also be completely multiplicative, i.e.
Dirichlet considered and as real numbers. Later on, as Riemann generalized to complex numbers for Euler's zeta function, giving the Riemann zeta function, other mathematicians will generalize and to complex numbers.
Euler product for Dirichlet L-functions
There is also an Euler product for Dirichlet L-functions
where is the th prime.
The Riemann zeta function is the special case obtained when is 1 for all .
The inverse of a Dirichlet L-function is obtained by Möbius inversion of its Dirichlet series
where is the Möbius function.