A note on \(r\)-gaps between zeros of the Riemann zeta-function (Q6570061)

Let \(N(T)\) be the counting function of the zeros \(\rho=\beta+ i\gamma\) of the Riemann zeta function \(\zeta(s)\) with \(0<\gamma<T\) counted with multiplicity. The Riemann-von Mangoldt formula yields \[N(T)\sim_{T \to \infty} \frac{T}{2\pi}\log T.\] Let consider the sequence of ordinates of zeros in the upper half-plane \(0<\gamma_{1}\leq\gamma_{2}\leq\cdots\leq\gamma_{n}\leq\gamma_{n+1}\leq\cdots\). The average gap between \(\gamma_{n+r}-\gamma_r\) is asymptotically \(2\pi r/\log\gamma_{n}\). In view of this, it is natural to define the normalized large/small \(r\)-gaps as \[ \lambda_r=\limsup_{n\to \infty}\frac{\gamma_{n+r}-\gamma_{n}}{2\pi r/\log\gamma_{n}} \quad \mathrm{and}\quad \mu_r=\liminf_{n\to \infty}\frac{\gamma_{n+r}-\gamma_{n}}{2\pi r/\log\gamma_{n}}.\] \textit{A. Selberg} [Collected papers. Volume I. Berlin etc.: Springer-Verlag (1989; Zbl 0675.10001)] and \textit{A. Fujii} [Bull. Am. Math. Soc. 81, 139--142 (1975; Zbl 0297.10026)] conjectured that \[\lambda_r>1+ \frac{\theta}{r^\alpha} \quad \mathrm{and}\quad \mu_r <1-\frac{\theta}{r^\alpha}\] for all positive intege \(r\) with \(\theta\) is an absolute positive constant, where \(\alpha\) may be taken as \(2/3\) unconditionally, and if we assume the Riemann Hypothesis as \(1/2\). However, they did not give any proof for those inequalities\N\NIn this paper under review, the author shows that there exist absolute positive constants \(\theta_1\) and \(\theta_2\) such that for any integer \(r\geq 1\) \[\lambda_r>1+ \frac{\theta_1}{r^\alpha} \quad \mathrm{and}\quad \mu_r<1-\frac{\theta_2}{r^\alpha},\] where \(\alpha\) may be taken as \(2/3\) unconditionally and as \(1/2\) under the Riemann Hypothesis.\N\NFinally, the explicit size of \(\theta_1\) and \(\theta_2\) is discussed uniformly for \(r\geq 1\) and assuming the Riemann hypothesis.