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

${\displaystyle L(s,\chi )\equiv \sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}={\frac {\chi (1)}{1^{s}}}+{\frac {\chi (2)}{2^{s}}}+{\frac {\chi (3)}{3^{s}}}+{\frac {\chi (4)}{4^{s}}}+\cdots ,\quad s>1,\,}$

where ${\displaystyle \scriptstyle \chi (n)\,}$ is the Dirichlet character of ${\displaystyle \scriptstyle n\,}$.

Dirichlet character

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

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

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

Euler product for Dirichlet L-functions

There is also an Euler product for Dirichlet L-functions

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

where ${\displaystyle \scriptstyle p_{i}\,}$ is the ${\displaystyle \scriptstyle i\,}$th prime.

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

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

${\displaystyle {\frac {1}{L(\chi ,s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)\chi (n)}{n^{s}}},\,}$

where ${\displaystyle \scriptstyle \mu (n)\,}$ is the Möbius function.