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On the success and failure of Gram’s law and the Rosser rule. (English) Zbl 1281.11084

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.

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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