The Riemann zeta function at the zeros of real-rooted trigonometric polynomials

Source abstract

Let $P$ be a trigonometric polynomial with only real zeros. We prove that at most $\frac{2}{3}$ of its zeros coincide with the zeros of the Riemann zeta function $ζ(s)$ on the critical line. We also show that $$\begin{equation*} \sup_{\substack{T\leq x\leq2T\\ P(x)=0}} \log \left|ζ\left(\frac{1}{2}+ix\right)\right| \geq (1+o(1))\sqrt{\frac{\log T}{6\log\log T}}. \end{equation*}$$ These results extend the corresponding theorems of Li and Radziwill on the zeros of $ζ(s)$ along vertical arithmetic progressions. The main novelty in our approach is the use of the work of Kurasov and Sarnak and Lee-Yang representations of Alon-Cohen-Vinzant, which show that the corresponding measure $\sum_{P(γ)=0} {\bf δ}_γ$ is crystalline.