The nonvanishing of certain character sums

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.