On the value distribution of the Riemann zeta-function
The author continues to investigate the properties of the Riemann zeta function \(\zeta(s)\) on the critical strip \(0\leq \mathrm{Re} s\leq 1\). It is known that the function \(S(t):=\frac{1}{\pi}\zeta(\frac 12+it)\) changes the sign infinitely often as \(t\to\infty\). In 1946, A. Selberg developed a method and applied it to obtain a series of new results on the value distribution of the Riemann zeta function. In particular, he established a lower bound for the number of points at which \(S(t)\) changes sign in an interval \((T, T+H]\) of length \(H\). In the paper under review, the author obtains refinements of the lower and upper bounds for \(M(T+H)-M(T)\), where \(M(T)\) denotes the number of sign changes of \(S(t)\) in the interval \((0,T]\).
scientific article; zbMATH DE number 5998392