CÁLCULO GRACELI SINTÉTICO [25]

$L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}.$ b $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

$\sum _{n=1}^{k}\chi (n)\exp(2\pi in/k).$ b $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

$L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}={\frac {1}{k^{s}}}\sum _{m=1}^{k}\chi (m)\;\zeta \left(s,{\frac {m}{k}}\right).$ b $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

$\zeta (s)={\frac {1}{k^{s}}}\sum _{m=1}^{k}\zeta \left(s,{\frac {m}{k}}\right).$ b $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

$\eta (s)=\left(1-2^{1-s}\right)\zeta (s)$ b $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

$\eta (s)=\sum _{n=1}^{\infty }{(-1)^{n-1} \over n^{s}}.$ b $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

$\zeta (s,q)=\sum _{k=0}^{\infty }(k+q)^{-s}.$ b $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

$\lim _{s\to 1}\left[\zeta (s,q)-{\frac {1}{s-1}}\right]={\frac {-\Gamma '(q)}{\Gamma (q)}}=-\psi (q)$ b $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

$\zeta (s,q)={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(q+k)^{1-s}.$ b $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

\zeta (1-s,x)={\frac {1}{2s}}\left[e^{-i\pi s/2}\beta (x;s)+e^{i\pi s/2}\beta (1-x;s)\right]

\beta (x;s)=2\Gamma (s+1)\sum _{n=1}^{\infty }{\frac {\exp(2\pi inx)}{(2\pi n)^{s}}}={\frac {2\Gamma (s+1)}{(2\pi )^{s}}}{\mbox{Li}}_{s}(e^{2\pi ix})

\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.

{ [nx]

n!

/ [pw] }

{\sqrt {\frac {2}{\pi }}}

({\frac {\pi }{2}})^{1/2}

{\frac {\pi }{2}}t^{2}

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

\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.

{ [nx]

n!

/ [pw] }

{\sqrt {\frac {2}{\pi }}}

({\frac {\pi }{2}})^{1/2}

{\frac {\pi }{2}}t^{2}

\zeta _{K}(s)=\sum _{I\subseteq O_{K}}(N_{K/\mathbb {Q} }(I))^{-s}

\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.

{ [nx]

n!

/ [pw] }

{\sqrt {\frac {2}{\pi }}}

({\frac {\pi }{2}})^{1/2}

{\frac {\pi }{2}}t^{2}

\zeta _{K}(s)=\prod _{P\subseteq O_{K}}{\frac {1}{1-(N_{K/\mathbb {Q} }(P))^{-s}}}.

\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.

{ [nx]

n!

/ [pw] }

{\sqrt {\frac {2}{\pi }}}

({\frac {\pi }{2}})^{1/2}

{\frac {\pi }{2}}t^{2}

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CÁLCULO GRACELI SINTÉTICO [25]

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