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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 \(L(s, \chi)\)
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