Dirichlet L-function

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

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

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

Dirichlet character

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

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

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

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

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 $\scriptstyle \mu (n)\,$ is the Möbius function.

See also

• Zeta functions
• Riemann zeta function