Let \(Z(t)\) be the Hardy function defined by \[ Z(t)=\left( \pi^{-it}\frac{\Gamma(\frac{1}{4}+\frac{it}{2})}{\Gamma(\frac{1}{4}-\frac{it}{2})}\right)^\frac{1}{2}\zeta\left(\frac{1}{2}+it\right), \] where \(\zeta(s)\) is the Riemann zeta function. Let \[ \Lambda^{(k)}=\lim\sup_{n \to \infty } \frac{t_{n+1}^{(k)}-t_{n}^{(k)}}{2\pi/\log t_n^{(k)}}, \] where \(\{t_n^{(k)}\}_{n\in \mathbb{N}}\) denotes the non-decreasing sequence of non-negative zeros of the \(k\)th derivative of \(Z(t)\).
In this paper under review, the authors prove the following main result \[ \Lambda^{(1)}>1.9. \] The above estimate is an improvement of a previous conditional result of J. B. Conrey and A. Ghosh [J. Lond. Math. Soc., II. Ser. 32, 193–202 (1985; Zbl 0582.10028)] and the second author’s unconditional estimate [R. R. Hall, Acta Arith. 111, No. 2, 125–140 (2004; Zbl 1153.11324)].
Assuming the Riemann hypothesis, the authors prove that there exist infinitely many zeros of \(Z(t)\) whose distances to the set of zeros of \(Z^{(2k)}(t)\) are small. As consequence, it is shown that for \(1\leq k\leq 3\) and any \(C > \log 4\), there are \[ \gg T \exp\left(-C\frac{\log T}{\log \log T} \right) \] ordinates \(t_n\in[T, 2T]\) whose distances to the set of zeros of \(Z^{(2k)}(t)\) are \(<\frac{2\pi}{\log T}\mu_k\) where \[ \mu_1=0.2 \qquad \mu_2=0.22 \qquad \mu_3=0.23. \]