A note on simple zeros of primitive Dirichlet \(L\)-functions (Q2786573)

Recall the generalized Riemann hypothesis (GRH) for Dirichlet \(L\)-functions: If \(\chi\) is a Dirichlet character, then the corresponding \(L\)-function \(L(s,\chi)\) has no zeros in the half-plane \(\mathrm{Re}(s) > 1/2\). A refinement of GRH, known as the \textit{simplicity hypothesis} asserts that not only are all non-trivial zeros on the line \(\mathrm{Re}(s) = 1/2\), but that they are also simple. Going back to celebrated work of \textit{H. L. Montgomery} [Proc. Sympos. Pure Math. 24, 181--193 (1973; Zbl 0268.10023)], the standard approach to the simplicity hypothesis has been to assume the GRH for the relevant \(L\)-functions and to study the simple zeros by means of estimates for so-called pair correlation functions. In the paper under review, the author studies the pair correlations of zeros of \(L\)-functions with primitive characters and deduces that, under the assumption of the GRH, at least 93.22.