On the zeros of the Riemann zeta function, lying in almost all short intervals of the critical line

Do Duc Tam

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 71-89

Résumé

In this paper, we study the distribution of non-trivial zeros of the Riemann zeta function $\zeta(s)$, which are on the critical line $\Re {s}=1/2$. On the half-plane $\Re{s}>1$, the Riemann zeta function is defined by Dirichlet series $$ \zeta(s)=\sum_{n=1}^{+\infty}n^{-s}, $$ and it can be analytically continued to the whole complex plane except the point $ s = 1$. It is well-known that the non-trivial zeros of the Riemann zeta function are symmetric about the real axis and the line $\Re{s}=1/2$. This line is called critical. In 1859, Riemann conjectured that all non-trivial zeros of the Riemann zeta function lie on the critical line $\Re{s}=1/2$. Hardy was the first to show in 1914 that $\zeta(1/2+it)$ has infinitely many real zeros. In 1942, Selberg obtained lower bound of the correct order of magnitude for the number zeros of the Riemann zeta functions on intervals of critical line $[T,T+H], H=T^{0.5+\varepsilon}$, where $\varepsilon $ — an arbitrary small constant. In 1984, A. A. Karatsuba proved Selberg's result for shorter intervals of critical line $[T,T+H], H=T^{27/82+\varepsilon}$. It is difficult to reduce the length of interval, which was pointed out above. However, if we consider this problem on average, then it was solved by Karatsuba. He proved that almost all intervals of line $ \Re {s} = 1/2 $ of the form $ [T, T + X^{\varepsilon}] $, where $ 0 $, contain more than $ c_0 (\varepsilon) T^{\varepsilon} \ln T $ zeros of odd orders of the function $ \zeta (1/2 + it) $. In 1988, Kicileva L. V. obtained result of this kind, but for the averaging intervals $ (X, X + X^{11/12 + \varepsilon}) $. In this paper, the length of the averaging interval has reduced. We proved Karatsuba's result for interval $ (X, X + X^{7/8 + \varepsilon})$. Bibliography: 17 titles.

Détail

Keywords: the Riemann zeta function, non-trivial zeros, critical line.