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

For any real

$$\beta _0\in [\tfrac{1}{2},1)$$

β 0 ∈ [ 1 2 , 1 ) , let

$$\textrm{GRH}[\beta _0]$$

GRH [ β 0 ] be the assertion that for every Dirichlet character

$$\chi$$

χ and all zeros

$$\rho =\beta +i\gamma$$

ρ = β + i γ of

$$L(s,\chi )$$

L ( s , χ ) , one has

$$\beta \leqslant \beta _0$$

β ⩽ β 0 (in particular,

$$\textrm{GRH}[\frac{1}{2}]$$

GRH [ 1 2 ] is the Generalized Riemann Hypothesis). In this paper, we show that the validity of

$$\textrm{GRH}[\frac{9}{10}]$$

GRH [ 9 10 ] depends only on certain distributional properties of the zeros of the Riemann zeta function

$$\zeta (s)$$

ζ ( s ) . No conditions are imposed on the zeros of nonprincipal Dirichlet L -functions.