$$\newcommand{acoloneqq}[0]{\ &\hspace-2pt\coloneqq}
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$$
定理
$N\ge2,n\ge1$
$(\eocase偶)\lor(\eocase奇)$
$\displaystyle L(n,\chi) =
-\frac1{2\cdot(n-1)!}\qty(-\frac\pi N)^n
\sum_{k=1}^{N-1} \chi(k)\cot^{(n-1)}\frac{\pi k}N$
補題
$\displaystyle \sum_{k=0}^\infty \qty[(z+k)^{-n}+(z-k-1)^{-n}] = -\frac{(-\pi)^n}{(n-1)!}\cot^{(n-1)}(\pi z)$
三角関数の部分分数展開
より、
$\displaystyle \pi\cot(\pi z) =
\lim_{j\to\infty} \sum_{|k|\le j} \frac1{z+k} =
\sum_{k=0}^\infty \qty(\frac1{z+k}+\frac1{z-k-1})$
これを$n-1$階微分する。
証明
$\beginend{align}{
\sahen &=
\sum_{j=0}^\infty \sum_{k=1}^{N-1}
\chi(k)(Nj+k)^{-n} \\&=
\frac12\sum_{j=0}^\infty \sum_{k=1}^{N-1}
\qty[\chi(k)(Nj+k)^{-n}+\chi(N-k)(Nj+N-k)^{-n}] \\&=
\frac12\sum_{j=0}^\infty \sum_{k=1}^{N-1}
\chi(k)\fqty[(Nj+k)^{-n}+(-1)^n(Nj+N-k)^{-n}]
\qbt{$n$と$\chi$の偶奇性の一致。} \\&=
\frac{N^{-n}}2\sum_{k=1}^{N-1} \chi(k)\sum_{j=0}^\infty
\qty[\qty(\frac kN+j)^{-n}+\qty(\frac kN-j-1)^{-n}] \\&=
\uhen \qbt{補題2}
}$
