On the zeros of the Riemann zeta function of large multiplicity
The author obtains a new upper bound for the number of zeros of the Riemann zeta function of a given multiplicity lying in a given rectangle of the critical strip thereby refining results of \textit{M. A. Korolev} [Izv. Math. 70, No. 3, 427--446 (2006); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 70, No. 3, 3--22 (2006; Zbl 1187.11027)]. Theorem 3. Suppose that \(\varepsilon\) is an arbitrarily small fixed number, \(0<\varepsilon<0.001\), \(T>T_0(\varepsilon)>0\) and \(H=T^{27/82+\varepsilon}\). Then, for any integer \(j\ge j_0\), the following estimate holds: \[ N_j(T+H)-N_j(T)\le \frac{2\kappa e^{3.1}}{(1-e^{-3})^2} (N(T+H)-N(T)) \exp(-\kappa j),\] where \(\kappa=2\varepsilon\sqrt{\varepsilon}/(3e\sqrt{B})\), \(B=e^{37}\pi^{-2}\), and \(j_0=(3+ 0.1\kappa+2(1.5\varepsilon e)^{2/3})/\kappa\). Theorem 4. For any \(j\ge 3.1/\alpha\) and \(T>T_0>0\), the following estimate holds: \[ N_j(T)\le beta N(T)\exp(-\alpha j),\] where \(\alpha=\frac{\pi\sqrt{5}}{3} 10^{-5}e^{-19.5}\), \(\beta=\frac{2\alpha e^{3.2}}{(1-e^{-3})^2}\).
scientific article; zbMATH DE number 5980809