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Gram’s law and the argument of the Riemann zeta function. (English) Zbl 1274.11132
This work belongs to the series of papers that the author has written on Gram’s law and related subjects [Dokl. Math. 67, No. 3, 396–397 (2003); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 390, No. 5, 588–589 (2003; Zbl 1234.11113)], [Izv. Math. 67, No. 2, 225–264 (2003); translation from Izv. Ross. Akad. Nauk Ser. Mat. 67, No. 2, 21–60 (2003; Zbl 1066.11038)], [Russ. Math. Surv. 61, No. 3, 389–482 (with A. A. Karatsuba) (2006; Zbl 1215.11081); translation from Usp. Mat. Nauk 61, No. 3, 3–92 (2006)], and [Izv. Math. 76, No. 2, 275–309 (2012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 2, 67-102 (2012; Zbl 1250.11081)]. If one writes the functional equation for \(\zeta(s)\) as \(\zeta(s)=\chi(s)\zeta(1-s)\) and defines \[ \vartheta(t)=-\frac1{2i}\log\chi(\tfrac12+it), \quad\text{ or }\quad e^{i\vartheta(t)}=\pi^{-it/2}\frac {\Gamma(\tfrac14+\tfrac12 it)}{| \Gamma(\tfrac14+\tfrac12 it)| }, \] then \(\vartheta(t)\in\mathbb R\) if \(t\in\mathbb R\). The Gram point \(t_n\) (\(n=0,1,2,\dots\)) is the unique solution of the equation \(\vartheta(t_n)=\pi(n-1)\). Gram points are investigated in detail and the function \(S(t)=\pi^{-1}\arg\zeta(\frac12+it)\) from the theory of the Riemann zeta-function \(\zeta(s)\), which is connected to them. For the calculation of \(\zeta(\frac12+it)\) the so-called Gram’s law is of importance. It states that the interval \((t_{n-1},t_n]\) contains a zero of Hardy’s function \[ Z(t):=\zeta(\tfrac12+it)\bigl(\chi(\tfrac12+it)\bigr)^{-1/2}, \] which is a real-valued, smooth function of \(t\). Although it is known that Gram’s law, in general, is not true, the topic is still of interest for the theory of \(Z(t)\) and \(\zeta(s)\).
A detailed discussion concerning Gram points and Gram’s law is given. Some interesting new statements concerning the behavior of the argument of the Riemann zeta function at the Gram points are proved. The results are applied to prove Selberg’s formulas connected with Gram’s Law.

MSC:
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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