## 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 $$L(s, \chi)$$

### Citations:

Zbl 0042.07901; Zbl 0061.08402
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