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Note on the trivial zeros of Dirichlet L-functions. (English) Zbl 0564.10044

If χ is a Dirichlet character mod k, then the Dirichlet L-function L(s,χ) has trivial zeros at negative integer points, i.e. χ(-1)=(-1) n implies L(-n,χ)=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,χ) by the Hurwitz zeta function ζ (s,a). The only property of ζ (s,a) he needs is proved by replacing z by -z in the contour integral of ζ (s,a).

Reviewer: H.Müller
MSC:
11M06ζ(s) and L(s,χ)