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For any real $\beta_0\in[\tfrac12,1)$, let ${\rm GRH}[\beta_0]$ be theassertion that for every Dirichlet character $\chi$ and all zeros$\rho=\beta+i\gamma$ of $L(s,\chi)$, one has $\beta\le\beta_0$ (in particular,${\rm GRH}[\frac12]$ is the Generalized Riemann Hypothesis). In this paper, weshow that the validity of ${\rm GRH}[\frac{9}{10}]$ depends only on certaindistributional properties of the zeros of the Riemann zeta function $\zeta(s)$.No conditions are imposed on the zeros of nonprincipal Dirichlet $L$-functions.

Towards the Generalized Riemann Hypothesis using only zeros of the Riemann zeta function