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How to show if $\chi_{0}$ is the trivial $\text{Dirichlet Character}$ then $$\lim_{\xi \to 1} (\xi -1) L(\xi,\chi_{0}) = \frac{\Phi(N)}{N}$$

where $\Phi$ is the $\text{Euler's Totient}$.

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up vote 5 down vote accepted

Here are some hints:

  • Use the product formula for $L(s, \chi) = \displaystyle\prod_{p} \biggl(1- \frac{\chi(p)}{p^s}\biggr)^{-1}$. Note $\chi_{0}(n) = \left\{\begin{array}{cc} 1 & p \nmid n \\\ 0 & p \mid n\end{array}\right.$

  • Use the formula $\Phi(n) = \displaystyle n \cdot \prod_{p} \biggl(1-\frac{1}{p}\biggr)$

  • Use the fact that $\displaystyle\lim_{z \to 1}\: (z-1)\:\zeta(z) =1$.

Complete solution:

See page 10 of the following link

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Thanks a lot for the answer. – Sarah Apr 17 '12 at 19:09

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