A variant of the Linnik–Sprindžuk theorem for simple zeros of Dirichlet L-functions

For a primitive Dirichlet character $X$, a new hypothesis ${\mathrm{RH}}_{\mathrm{sim}}^{†}\left[X\right]$ is introduced, which asserts that (1) all simple zeros of $L(s,X)$ in the critical strip are located on the critical line, and (2) these zeros satisfy some specific conditions on their vertical distribution. We show that ${\mathrm{RH}}_{\mathrm{sim}}^{†}\left[X\right]$ (for any $X$) follows from the generalized Riemann hypothesis.

Assuming only the generalized Lindelöf hypothesis, we show that if ${\mathrm{RH}}_{\mathrm{sim}}^{†}\left[X\right]$ holds for one primitive character $X$, then it holds for every such $X$. If this occurs, then for every character $\chi$ (primitive or not), all simple zeros of $L(s,\chi )$ in the critical strip are located on the critical line. In particular, Siegel zeros cannot exist in this situation.