On the distribution of values of the argument of the Riemann zeta-function

• Published in: Journal of number theory Vol. 200; pp. 96 - 131
• Main Authors: Ivić, Aleksandar P., Korolev, Maxim A.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.07.2019
• Subjects: Argument of Riemann zeta-function; Critical line; Density theorem; Hardy's function; Hermite polynomials; Riemann zeta-function; Value distribution; Density theorem; 11M06; Value distribution; Hardy's function; Critical line; Hermite polynomials; Riemann zeta-function; Argument of Riemann zeta-function
• ISSN: 0022-314X; 1096-1658
• DOI: 10.1016/j.jnt.2018.12.011

Let S(t):=1πarg⁡ζ(12+it). We prove that, for T27/82+ε⩽H⩽T, we havemes{t∈[T,T+H]:S(t)>0}=H2+O(Hlog3⁡Tεlog2⁡T), where the O-constant is absolute. A similar formula holds for the measure of the set with S(t)<0, where logk⁡T=log⁡(logk−1⁡T). This result is derived from an asymptotic formula for the distribution of values of S(t), which is uniform in the relevant parameters, and this is of crucial importance. This in fact depends on the distribution of values of the Dirichlet polynomial which approximates S(t), namely (p denotes primes)Vy(t)=∑p⩽ysin⁡(tlog⁡p)p.