![Pending changes are displayed on this page Pending changes are displayed on this page](/w/extensions/FlaggedRevs/frontend/modules/img/1.png)
There are no approved revisions of this page, so it may
not have been
reviewed.
This article page is a stub, please help by expanding it.
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
)
![{\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,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/90476c8731d2eeb2033fa7dde9aaad0ee320cb93)
where
is the Dirichlet character of
.
Dirichlet character
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
![{\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,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e87f832ed22efa9a1bae2bbcb527fd45e7d4d6de)
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
![{\displaystyle {\frac {1}{L(\chi ,s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)\chi (n)}{n^{s}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/dc6defe1f7d018d81f03d3d666cd9c8ae27bf36b)
where
is the Möbius function.
See also