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How would I calculate a function such as $$\sum_{n \leq x}r(n) = L(1, \chi) \cdot x + O(x^{1-\eta}),$$ where $r(n) = \sum_{d|n} \chi(d)$?

The part I'm having difficulty calculating is the L-function part. This particular L-function is equal to $$\sum_{n = 1}^{\infty} \frac{\chi(n)}{n},$$ where $\chi$ is a Dirichlet character. How would you evaluate this function? I looked on the Wikipedia page for both Dirichlet L-Series and Dirichlet Character, and neither had an explicit way of computing the summation. But the Dirichlet character page said that it could be calculated if one knew the modulus of the Dirichlet character? As far as I know, I don't the the modulus for this Dirichlet character. Is knowing the modulus of the character imperative?

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  • $\begingroup$ I only know the formula with the conductor: en.wikipedia.org/wiki/… . $\endgroup$
    – Asvin
    Nov 29, 2017 at 9:44
  • $\begingroup$ Your question is slightly ambiguous: how are you given $\chi$ if you don't know its modulus? Anyway, the following Pari/GP script by Henri Cohen computes numerically Dirichlet L-series using the Hurwitz zeta function : pari.math.u-bordeaux.fr/Scripts/cohen.gp $\endgroup$ Nov 29, 2017 at 9:54
  • $\begingroup$ If $\chi$ is a nontrivial character with modulus $m$ then $L(\chi,1)=(-1/m) \sum_{a=1}^m \chi(a) \Gamma'(a/m)/\Gamma(a/m)$. See for instance "An introduction to Zeta Functions" by P. Cartier in the volume "From Number Theory to Physics", Chapter 1. $\endgroup$ Nov 29, 2017 at 10:21

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If $\chi$ is $q$-periodic and zero mean then $$L(1,\chi) =\sum_{n=1}^\infty \frac{\chi(n)}{n}= \sum_{k=1}^{q-1} \frac{\hat{\chi}(k)}{q} \sum_{n=1}^\infty \frac{e^{2i \pi nk/q}}{n}=-\sum_{k=1}^{q-1} \frac{\hat{\chi}(k)}{q} \log(1-e^{2i \pi k/q})$$ where $\hat{\chi}(k) = \sum_{n=1}^q \chi(n) e^{-2i \pi nk/q}$. If $\chi$ is a primitive character then $\hat{\chi}(k) = \overline{\chi(k)} \hat{\chi}(1)$

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