On the primitive of the Hardy function \(Z(t)\)
type: Item
instance of: scholarly article
title: On the primitive of the Hardy function \(Z(t)\)
zbMATH Open document ID: 1162.11045
DOI: 10.1134/S1064562407020317
author: M. A. Korolev
published in: Doklady Mathematics
publication date: 2009-01-19T00:00:00Z
review text: 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 \(F(T)\).
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 5493407
zbMATH Keywords: Hardy function \(Z(t)\)
zbMATH Keywords: zeros
zbMATH Keywords: exponential sums
MaRDI profile type: Publication
cites work: The Riemann zeta-function. Transl. from the Russian by Neal Koblitz
cites work: Q5621389
cites work: On the integral of Hardy's function
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description: scientific article; zbMATH DE number 5493407