Trudgian, Timothy On the success and failure of Gram’s law and the Rosser rule. (English) Zbl 1281.11084 Acta Arith. 148, No. 3, 225-256 (2011). Let \(\zeta(s)\) be the Riemann zeta function and let \(g_\nu\), \(\nu=-1, 0, 1, \dots\) be Gram points. This interesting paper is devoted to known and new results concerning Gram Law, the Weak Gram Law, and Rosser’s Rule. It is studied whether these phenomenons are true or false for infinitely many intervals, and for a positive proportion of intervals.As an example we formulate the following result. Gram Law is said to hold if, given Gram points \(g_\nu\) and \(g_{\nu+1}\), there is exactly one zero of \(\zeta(\frac12+it)\) for some \(t\) in the interval \((g_\nu, g_{\nu+1}]\). The author shows that for sufficiently large \(T\) there is a positive proportion of Gram intervals between \(T\) and \(2T\) which do not contain a zero of \(\zeta(s)\). Thus there is a positive proportion of failures of the Weak Gram Law.The paper is based on the author’s PhD thesis. Reviewer: Ramūnas Garunkštis (Vilnius) Cited in 13 Documents MSC: 11M06 \(\zeta (s)\) and \(L(s, \chi)\) 11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses Keywords:Riemann zeta-function; zeros on the critical line; Gram’s law; Rosser’s rule PDFBibTeX XMLCite \textit{T. Trudgian}, Acta Arith. 148, No. 3, 225--256 (2011; Zbl 1281.11084) Full Text: DOI Link Online Encyclopedia of Integer Sequences: Indices n of (”bad”) Gram points g(n) for which (-1)^n Z(g(n)) < 0, where Z(t) is the Riemann-Siegel Z-function. Violations of Rosser’s rule: numbers n such that the Gram block [ g(n), g(n+k) ) contains fewer than k points t such that Z(t) = 0, where Z(t) is the Riemann-Siegel Z-function.