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Note on the trivial zeros of Dirichlet L-functions. (English) Zbl 0564.10044
If $$\chi$$ is a Dirichlet character mod k, then the Dirichlet L-function L(s,$$\chi)$$ has trivial zeros at negative integer points, i.e. $$\chi (- 1)=(-1)^ n$$ implies $$L(-n,\chi)=0$$. Usually this result is proved with the aid of the functional equation for the L-function. As the functional equation is only valid for primitive characters, some additional arguments are necessary.
In this note the author gives a very short proof using the representation of L(s,$$\chi)$$ by the Hurwitz zeta function $$\zeta$$ (s,a). The only property of $$\zeta$$ (s,a) he needs is proved by replacing z by -z in the contour integral of $$\zeta$$ (s,a).
Reviewer: H.Müller
##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
##### Keywords:
Hurwitz zeta function
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