On new results related to Gram's law (Q2866961)

For an integer \(n\) let \(t_n\) be the \textit{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 \textit{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 \(o(N)\).NEWLINENEWLINEIn 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 \(50 \%\).NEWLINENEWLINEFurther, 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.