Dense clusters of zeros near the zero-free region of \(\zeta(s)\) (Q6540616)

Let us recall the Korobov and Vinogradov zero-free region for the Riemann zeta function \(\zeta(s)\): \N\[\N\sigma > 1 -\frac{ c}{(\log \tau)^{2/3}(\log\log \tau)^{1/3}}\ \ (\tau:= |t| + 100).\N\]\NLet \(l(\tau):=(\log \tau)^{2/3}(\log\log \tau)^{1/3}\) (\(\tau\geq100\)). For any positive real-valued function \(f\), let \(R_{f}\) be the region consisting of complex number \(s=\sigma+it\) such that \[\sigma>1-\frac{f(\tau)}{l(\tau)},\] where \(\tau:=|t|+100 \ \ (t\in{\mathbb{R}})\).\N\NIn this paper under review, the author proves in Theorem 1.1 that there exist infinitely many distinct dense clusters of zeros of \(\zeta(s)\) lying close to the edge of the zero-free region. Similar results for \(L\)-functions associated to nonquadratic Dirichlet characters \(\chi\) modulo \(q\geq 2\) are given in Theorem 1.2.