Dirichlet L-functions

<< Lerch's Transcendent

Dirichlet L-functions

Mathematical Applications >>

Definitions and Basic Properties

${\displaystyle{\displaystyle{\displaystyle L\left(s,\chi\right)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\chi(n)}}}$ is a Dirichlet character~${\displaystyle{\displaystyle{\displaystyle\pmod{k}}}}$

${\displaystyle{\displaystyle{\displaystyle L\left(s,\chi\right)=\prod_{p}\left(1-\frac{\chi(p)}{p^{s}}\right)^{-1}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$

${\displaystyle{\displaystyle{\displaystyle L\left(s,\chi\right)=k^{-s}\sum_{r=1}^{k-1}\chi(r)\HurwitzZeta@{s}{\frac{r}{k}}}}}$

Constraint(s): hold for all ${\displaystyle{\displaystyle{\displaystyle s}}}$ if ${\displaystyle{\displaystyle{\displaystyle\chi\neq\chi_{1}}}}$, and for all ${\displaystyle{\displaystyle{\displaystyle s}}}$ (${\displaystyle{\displaystyle{\displaystyle\neq 1}}}$) if ${\displaystyle{\displaystyle{\displaystyle\chi=\chi_{1}}}}$

${\displaystyle{\displaystyle{\displaystyle L\left(s,\chi\right)=L\left(s,\chi_{0}\right)\prod_{p\mathbin{|}k}\left(1-\frac{\chi_{0}(p)}{p^{s}}\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\chi_{0}}}}$ is a primitive character (mod ${\displaystyle{\displaystyle{\displaystyle d}}}$) for some positive divisor ${\displaystyle{\displaystyle{\displaystyle d}}}$ of ${\displaystyle{\displaystyle{\displaystyle k}}}$ &
hold for all ${\displaystyle{\displaystyle{\displaystyle s}}}$ if ${\displaystyle{\displaystyle{\displaystyle\chi\neq\chi_{1}}}}$, and for all ${\displaystyle{\displaystyle{\displaystyle s}}}$ (${\displaystyle{\displaystyle{\displaystyle\neq 1}}}$) if ${\displaystyle{\displaystyle{\displaystyle\chi=\chi_{1}}}}$

${\displaystyle{\displaystyle{\displaystyle L\left(1-s,\chi\right)=\frac{k^{s-1}\Gamma\left(s\right)}{(2\pi)^{s}}\*{\left({\mathrm{e}^{-\pi\mathrm{i}s/2}}+\chi(-1){\mathrm{e}^{\pi\mathrm{i}s/2}}\right)}\*G(\chi)L\left(s,\overline{\chi}\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle{\displaystyle G(\chi)=\sum_{r=1}^{k}\chi(r){\mathrm{e}^{2\pi\mathrm{i}r/k}}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\chi}}}$ is a primitive character (mod ${\displaystyle{\displaystyle{\displaystyle k}}}$) &
${\displaystyle{\displaystyle{\displaystyle\overline{\chi}}}}$ is the complex conjugate of ${\displaystyle{\displaystyle{\displaystyle\chi}}}$

Zeros

${\displaystyle{\displaystyle{\displaystyle L\left(-2n,\chi\right)=0\text{\quad if\quad}\chi(-1)=1}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\dots}}}$

${\displaystyle{\displaystyle{\displaystyle L\left(-2n-1,\chi\right)=0\text{\quad if\quad}\chi(-1)=-1}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\dots}}}$

${\displaystyle{\displaystyle{\displaystyle L\left(1,\chi\right)\neq 0\text{\quad if\quad}\chi\neq-1}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\chi_{1}}}}$ is the principal character~${\displaystyle{\displaystyle{\displaystyle\pmod{k}}}}$

${\displaystyle{\displaystyle{\displaystyle L\left(0,\chi\right)=\begin{cases}\displaystyle-\frac{1}{k}\sum_{r=1}^{k}r\chi(r),&\chi\neq\chi_{1},\\ 0,&\chi=\chi_{1}.\end{cases}}}}$

<< Lerch's Transcendent

Dirichlet L-functions

Mathematical Applications >>