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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,$\chi \right)$ has trivial zeros at negative integer points, i.e. $\chi \left(-1\right)={\left(-1\right)}^{n}$ implies $L\left(-n,\chi \right)=0$. 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,$\chi \right)$ 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 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$
##### Keywords:
Hurwitz zeta function