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

Apostol, Tom M.

Proc. Am. Math. Soc. 94, 29-30 (1985).

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