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# Dirichlet L-function

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 $\{a,\,a+k,\,a+2k,\,a+3k,\,\ldots \}\,$ , where $a\,$ and $k\,$ are coprime, there are infinitely many primes (i.e. there are infinitely many primes for each residue class $a\,$ coprime to $k\,$ )

$L(s,\chi )\equiv \sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}={\frac {\chi (1)}{1^{s}}}+{\frac {\chi (2)}{2^{s}}}+{\frac {\chi (3)}{3^{s}}}+{\frac {\chi (4)}{4^{s}}}+\cdots ,\quad s>1,\,$ where $\chi (n)\,$ is the Dirichlet character of $n\,$ .

## Dirichlet character

The Dirichlet character $\chi (n)\,$ is a function of $n{\bmod {k}}\,$ , i.e. it is a function of the remainder (residue class) of $n\,$ when divided by $k\,$ and it must be 0 when $n\,$ and $k\,$ are not coprime. It must also be completely multiplicative, i.e.

$\chi (mn)=\chi (m)\cdot \chi (n)\,$ .

Dirichlet considered $s\,$ and $\chi (n)\,$ as real numbers. Later on, as Riemann generalized $s\,$ to complex numbers for Euler's zeta function, giving the Riemann zeta function, other mathematicians will generalize $s\,$ and $\chi (n)\,$ to complex numbers.

## Euler product for Dirichlet L-functions

There is also an Euler product for Dirichlet L-functions

$L(s,\chi )=\prod _{i=1}^{\infty }{\frac {1}{1-{\tfrac {\chi (p_{i})}{{p_{i}}^{s}}}}}={\frac {1}{1-{\tfrac {\chi (2)}{2^{s}}}}}\cdot {\frac {1}{1-{\tfrac {\chi (3)}{3^{s}}}}}\cdot {\frac {1}{1-{\tfrac {\chi (5)}{5^{s}}}}}\cdot {\frac {1}{1-{\tfrac {\chi (7)}{7^{s}}}}}\cdot {\frac {1}{1-{\tfrac {\chi (11)}{11^{s}}}}}\cdots ,\quad \sigma ={\mathfrak {R}}(s)>0,\,$ where $p_{i}\,$ is the $i\,$ th prime.

The Riemann zeta function is the special case obtained when $\chi (n)\,$ is 1 for all $n\,$ .

The inverse of a Dirichlet L-function is obtained by Möbius inversion of its Dirichlet series

${\frac {1}{L(\chi ,s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)\chi (n)}{n^{s}}},\,$ where $\mu (n)\,$ is the Möbius function.