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.
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 45 (1974), 164-166
- MSC: Primary 12A55; Secondary 10G05, 12A35
- DOI: https://doi.org/10.1090/S0002-9939-1974-0354611-7
- MathSciNet review: 0354611