The Generalized Riemann Hypothesis from zeros of a single L-function

Abstract

For each primitive Dirichlet character $\chi$, a hypothesis ${\mathrm{GRH}}^{†}\left[\chi \right]$ is formulated in terms of zeros of the associated $L$-function $L(s,\chi )$. It is shown that for any such character, ${\mathrm{GRH}}^{†}\left[\chi \right]$ is equivalent to the Generalized Riemann Hypothesis.