勒奇超越函数

赫尔维茨ζ函数。当勒奇函数中的z=1时，化为赫尔维茨ζ函数：

${\displaystyle L(1,s,a)=\zeta (s,a)}$

多重对数函数,当勒奇函数中a=1,则化为多重对数函数

${\displaystyle L(z,s,1)=Li_{s}(z)}$

勒让德χ函数可以用勒奇超越函数表示,

${\displaystyle \,\chi _{n}(z)=2^{-n}z\Phi (z^{2},n,1/2).}$

作为赫尔维茨ζ函数的特例，黎曼ζ函数可以表示为

${\displaystyle \,\zeta (s)=\Phi (1,s,1).}$

狄利克雷η函数可以表示为

${\displaystyle \,\eta (s)=\Phi (-1,s,1).}$

${\displaystyle L(z,s,a)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {z^{x}}{(a+x)^{s}}}dx}$

${\displaystyle \Phi (z,s,q)={\frac {1}{1-z}}\sum _{n=0}^{\infty }\left({\frac {-z}{1-z}}\right)^{n}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(q+k)^{-s}.}$

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