On the number of sign changes of the function \(S(t)\) on a short interval. (English. Russian original) Zbl 1145.11323

Korolev, M. A.

Dokl. Math. 65, No. 1, 58-59 (2002); translation from Dokl. Akad. Nauk 382, No. 4, 446-447 (2002).

For a real number \(t\) different from the imaginary part of a zero of the Riemann zeta function \(\zeta(s)\),
\[ S(t)= \tfrac1\pi\arg \zeta\bigl(\tfrac12+it\bigr), \]
where \(\arg \zeta(\frac12+it)\) is obtained by a continuous continuation along the polygonal line joining the point \(s=2\), \(s=2+it\), and \(s=\frac12+it\) of the branch of the argument of \(\zeta(s)\) for which \(\arg\zeta(2)=0\).
Let \(N_1(T)\) be the number of sign changes of \(S(t)\) on the interval \((0;T]\).
Using the results of A. Selberg [Arch. Math. Naturvid. 48, No. 5, 89–155 (1946; Zbl 0061.08402)], A. Ghosh [in: Recent progress in analytic number theory, Symp. Durham 1979, Vol. 1, 25–46 (1981; Zbl 0457.10018)] and A. A. Karatsuba [Izv. Math. 60, No. 5, 901–931 (1996); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 60, No. 5, 27–56 (1996; Zbl 0898.11032)], this author proved the following assertion.
Theorem. Suppose that \(0<\alpha<0.001\) is an arbitrary fixed number and \(H=T^{\frac{27}{82}+\alpha}\). Then there exist positive constants \(T_1= T_1(\alpha)\) and \(c_1=c_1(\alpha)\) such that, at \(T\geq T_1\), we have
\[ N_1(T+H)-N_1(T)\geq H\ln T\exp \biggl(- \frac{c\ln\ln T}{\sqrt{\ln\ln\ln T}} \biggr). \]
Corollary. There exist absolute positive constants \(T_0\) and \(c_0\) such that, at \(T\geq T_0\), we have
\[ N_1(T)\geq T\ln T\exp\biggl(- \frac{c_0\ln\ln T}{\sqrt{\ln\ln\ln T}}\biggr). \]

Reviewer: Olaf Ninnemann (Berlin)

MSC:

11M06

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

Citations:

Zbl 0061.08402; Zbl 0457.10018; Zbl 0898.11032