The nonvanishing of certain character sums
Author: S. Ullom
Journal: Proc. Amer. Math. Soc. 45 (1974), 164-166
MSC: Primary 12A55; Secondary 10G05, 12A35
MathSciNet review: 0354611
Full-text PDF Free Access
Abstract: Let $\chi$ be a Dirichlet character with conductor $f$ and $M(\chi ) = \Sigma a\bar \chi (a)$, summation over integers $a$ prime to $f$ and $1 \leqslant a < f$. It is well known that the nonvanishing of the Dirichlet $L$-function $L(s,\chi )$ at $s = 1$ implies $M(\chi ) \ne 0$ for $\chi$ imaginary, i.e. $\chi ( - 1) = - 1$. This article provides a purely algebraic proof that $M(\chi ) \ne 0$ when the conductor $f$ is a prime power and the imaginary $\chi$ is either a faithful character or has order a power of 2.
H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952. MR 14, 141.