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Some \(\Omega\)-theorems for the Riemann zeta-function. (English) Zbl 0606.10031

Several new Omega results involving the functions \[ S(T)=(1/\pi) \arg \zeta (1/2+iT),\quad S_ 1(T)=\int^{T}_{0}S(t) dt \] are proved. As is well-known [see Chapter IX of E. C. Titchmarsh, The theory of the Riemann zeta-function (1951; Zbl 0042.07901)] these functions are closely related to the distribution of the imaginary parts of the zeros of \(\zeta\) (s). A. Selberg [Arch. Math. Naturvid. 48, No.5, 89-155 (1946; Zbl 0061.08402)] has proved \[ S(T)=\Omega_{\pm}((\log T)^{1/3}\quad (\log \log T)^{-7/3}), \]
\[ S_ 1(T)=\Omega_+((\log T)^{1/2}\quad (\log \log T)^{-4}),\quad S_ 1(T)=\Omega_-((\log T)^{1/3}\quad (\log \log T)^{-10/3}). \] The author refines Selberg’s arguments and proves now, among other things, \[ S(T)=\Omega_{\pm}((\log T/\log \log T)^{1/3}), \]
\[ S_ 1(T)=\Omega_+((\log T)^{1/2}\quad (\log \log T)^{-9/4}),\quad S_ 1(T)=\Omega_-((\log T)^{1/3}\quad (\log \log T)^{-4/3}), \] and if one assumes the Riemann hypothesis, then \[ S_ 1(T)=\Omega_{\pm}((\log T)^{1/2}\quad (\log \log T)^{-3/2}). \] It may be conjectured that both S(T) and \(S_ 1(T)\) are of the order \((\log T)^{1/2+o(1)}\) as \(T\to \infty\), although it is known e.g. only that \(S(T)=O(\log T)\) (O(log T/log log T) if the RH holds), so that there is still a considerable gap between O- and \(\Omega\)-results. Nevertheless, the author’s results represent an important contribution to the study of S(T) and \(S_ 1(T)\).
Reviewer: A.Ivić

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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