On the zeros of higher derivatives of Hardy's \(Z\)-function
The function in question is \[ Z(t) := \chi^{-1/2}({\textstyle{1\over 2}} + it)\zeta({\textstyle{1\over 2}} + it),\quad \chi(s) = \zeta(s)/\zeta(1-s) = 2^s\pi^{s-1}\sin\left({\pi s\over 2}\right) \Gamma(s), \] so that \(Z(t)\) is real for real \(t\) and \(|Z(t)|= |\zeta({1\over 2}+it)|\). Hardy's function \(Z(t)\) is especially useful in investigations of zeros of the Riemann zeta-function on the critical line \(\Re s = {1\over 2}\). \textit{R. J. Anderson} [J. Math. Anal. Appl. 118, 323-340 (1986; Zbl 0592.10035)] proved that the number of zeros of \(Z'(t)\) in \((0, T] \) is \[ \leq {T\over 2\pi}\log{T\over 2\pi} - {T\over 2\pi} + O(\log T), \] with equality holding if the Riemann Hypothesis (that all complex zeros of \(\zeta(s)\) have real parts equal to \({1\over 2}\)) is true. The present authors successfully generalize Anderson's method and prove that the number of zeros of \(Z^{(k)}(t)\) in \((0, T] \) is \[ \leq {T\over 2\pi}\log{T\over 2\pi} - {T\over 2\pi} + O_k(\log T),\leqno(1) \] with equality holding if the Riemann Hypothesis is true. To this end they introduce the meromorphic function \(\eta_k(s)\) for each \(k\), precisely defined in the text, whose zeros on \(\Re s = {1\over 2}\) essentially coincide with the zeros of \(Z^{(k)}(t)\). They prove that the number of zeros of \(\eta_k(s)\) in a suitable rectangle is given by the expression in (1).
higher derivatives of Hardy's \(Z\)-function
scientific article; zbMATH DE number 1275046
scientific article