Selberg's method and the multiplicities of the zeroes of the Riemann zeta-function (Q2910127)

If \(T\) is not the ordinate of a zero of the Riemann zeta-function \(\zeta(s)\), let \(S(T)\) denote the value of \(\pi^{-1}\arg\zeta(\frac{1}{2}+iT)\). Otherwise we define \(S(T)\) as \(\lim_{t\to T^+}S(t)\). There is an intimate connection between \(S(T)\) and the number of non-trivial zeros of \(\zeta(s)\) or the multiplicity of these zeros. For example, it is well-known that \(S(T)\ll \log T\) implies that the maximal multiplicity \(M(T)\) of zeros between \(T\) and \(2T\) satisfies \(M(T)\ll \log T\). Moreover, under the Riemann hypothesis, the following implication holds: NEWLINE\[NEWLINE S(T)\ll\frac{\log T}{\log\log T}\Longrightarrow M(T)\ll \frac{\log T}{\log\log T}. NEWLINE\]NEWLINE In the paper under review the author introduces the reasoning how the best unconditional bound for \(M(T)\) can be obtained by estimating the constant \(c(m)\) in the following theorem due to Selberg.NEWLINENEWLINEIf \(T^\alpha<H\leq T\), where \(\alpha\in(\frac{1}{2},1]\) is fixed, and \(T^{\frac{\alpha-\frac{1}{2}}{m}}\leq x\leq H^{\frac{1}{m}}\) for \(m\in\mathbb{Z}^+\), then NEWLINE\[NEWLINE \int_T^{T+H}\left|S(t)+\frac{1}{\pi}\sum_{p<x}\frac{\sin(t\log p)}{\sqrt{p}}\right|^{2m}dt\leq c(m)H, NEWLINE\]NEWLINE where \(c(m)\) depends only on \(m\).NEWLINENEWLINEMore precisely, the author explains how the inequality \(c(m)\leq (Am)^{\alpha m}\) implies that \(M(T)\ll (\log T)^{\alpha/2}\). Comparing this with Tsang's result (\(c(m)\leq (Am)^{2m}\)) gives the best bound for \(M(T)\).