Korolëv, M. A. On new results related to Gram’s law. (English. Russian original) Zbl 1303.11100 Izv. Math. 77, No. 5, 917-940 (2013); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 5, 71-94 (2013). For an integer \(n\) let \(t_n\) be the Gram point of the Riemann zeta-function \(\zeta(s)\). Let \(0<\gamma_1<\dots\leq\gamma_n\leq\dots\) be the ordinates of the zeros of \(\zeta(s)\). For a given ordinate \(\gamma_n\) define the number \(m=m(n)\) such that the inequality \(t_{m-1}<\gamma_n\leq t_{m}\) holds. Let \(\Delta_n=m-n\). It is said that \(\gamma_n\) satisfies Gram’s law if and only if \(\Delta_n=0\). It is known that the numbers of both positive and negative terms among the first \(N\) terms of the sequence \(\Delta_n\) are asymptotically equivalent to \(N/2\) as \(N\to\infty\), while the number of indices \(n\leq N\) such that \(\Delta_n=0\) is \(o(N)\).In this paper the distribution of signs of the quantities \(\Delta_n, \Delta_{n+1},\dots,\Delta_{n+k-1}\) is investigated. For example, if \(k=1\) then the indices \(n\) (for which \(\Delta_n\Delta_{n+1}\neq0\)) can be divided into three classes depending on which of the following conditions hold: 1) \(\Delta_n>0\), \(\Delta_{n+1}<0\); 2) \(\Delta_n<0\), \(\Delta_{n+1}<0\); 3) \(\Delta_n<0\), \(\Delta_{n+1}>0\). Note that the case \(\Delta_n>0\), \(\Delta_{n+1}<0\) is impossible. The author shows that the proportions of the number of elements in the first, second, and third classes are \(50 \%\).Further, the author considers the value of the product \(|\Delta_n\cdots\Delta_{n+k-1}|\). For a precise formulations of the obtained results see Theorems 1 and 2. Reviewer: Ramūnas Garunkštis (Vilnius) Cited in 1 ReviewCited in 4 Documents MSC: 11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses 11M06 \(\zeta (s)\) and \(L(s, \chi)\) Keywords:Riemann zeta-function; Gram’s law; Gram’s rule PDFBibTeX XMLCite \textit{M. A. Korolëv}, Izv. Math. 77, No. 5, 917--940 (2013; Zbl 1303.11100); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 5, 71--94 (2013) Full Text: DOI References: [1] A. Selberg, “The zeta-function and the Riemann hypothesis”, C. R. Dixieme Congre\?s Math. Scandinaves 1946, Jul. Gjellerups Forlag, Copenhagen, 1947, 187 – 200 · Zbl 0030.05003 [2] A. Selberg, Collected papers, v. I, Springer-Verlag, Berlin, 1989 · Zbl 0675.10001 [3] М. А. Королe\"в, “Закон Грама и гипотеза Сельберга о распределении нулей дзета-функции Римана”, Изв. РАН. Сер. матем., 74:4 (2010), 83 – 118 · Zbl 1257.11080 [4] M. A. Korolev, “Gram/s law and Selberg/s conjecture on the distribution of zeros of the Riemann zeta function”, Izv. Math., 74:4 (2010), 743 – 780 · Zbl 1257.11080 [5] М. А. Королe\"в, “О законе Грама в теории дзета-функции Римана”, Изв. РАН. Сер. матем., 76:2 (2012), 67 – 102 · Zbl 1250.11081 [6] M. A. Korolev, “On Gram/s law in the theory of the Riemann zeta-function”, Izv. Math., 76:2 (2012), 275 – 309 · Zbl 1250.11081 [7] М. А. Королe\"в, “О больших расстояниях между соседними нулями дзета-функции Римана”, Изв. РАН. Сер. матем., 72:2 (2008), 91 – 104 · Zbl 1180.11030 [8] M. A. Korolev, “On large distances between consecutive zeros of the Riemann zeta-function”, Izv. Math., 72:2 (2008), 291 – 304 · Zbl 1180.11030 [9] A. Ghosh, “On the Riemann zeta-function – mean value theorems and the distribution of \(|S(t)|\)”, J. Number Theory, 17:1 (1983), 93 – 102 · Zbl 0511.10030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.