The Generalized Riemann Hypothesis for Dirichlet L-functions

Let $\chi$ be a Dirichlet character modulo $q greater than or equal to 1$, and let ${\mathcal N}(\frac{1}{2},\infty,\chi)$ denote the number of nontrivial zeros of the associated Dirichlet $L$-function whose real parts are at least $\frac{1}{2}$. Generalized Riemann Hypothesis. For every Dirichlet character $\chi$ modulo $q$,

$$\mathcal N\bigg(\frac{1}{2},\infty,\chi\bigg)=0.$$

Equivalently, all nontrivial zeros of every Dirichlet $L$-function lie on the critical line. This is a standard major open problem and is used in the paper as a hypothesis under which the restricted-prime Goldbach results obtain shorter intervals.