On the primitive of the Hardy function \(Z(t)\) (Q960712)

The author presents (without proof) new results on \(F(T) = \int_{2\pi}^TZ(t)\,dt\), where (\(t\) is real) \[ Z(t) := \zeta({1\over2}+it)\chi^{-1/2}({1\over2}+it), \quad \zeta(s) = \chi(s)\zeta(1-s)\quad(\forall s = \sigma+it) \] is Hardy's function. This function has zeros at the points where \(\zeta(s)\) has zeros on the ``critical line'' \(\sigma = 1/2\), and \(|Z(t)| = |\zeta({1\over2}+it)|\), hence it is an important tool in the investigation of zeta-zeros. The reviewer [Arch. Math. 83, 41--47 (2004; Zbl 1168.11319)] proved that \(F(T) = O(T^{1/4+\varepsilon})\) for any given \(\varepsilon>0\) and conjectured that \(F(T) = \Omega_\pm(T^{1/4})\). The present author announces that \[ |F(T)| <18.2\,T^{1/4}\quad(T\geq T_0), \quad F(X_n) < -X_n^{1/4},\; F(Y_n) > Y_n^{1/4} \] for two explicitly constructed sequences \(X_n, Y_n\) tending to \(+\infty\) when \(n\to\infty\) (there is a misprint on p. 298 where \(-Y_\nu^{1/4}\) should be \(Y_\nu^{1/4}\)). These remarkable results sharpen the reviewer's bound, establish his conjecture and thus establish the true order of \(F(T)\).