On real zeros of the derivative of the Hardy function

Sh. A. Khayrulloev

Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 234-240

Résumé

The existence of the zeros of the Riemann zeta-function in the short segments of the critical line (or the real zeros of Hardy's function $Z(t)$, that is the same) is one of the topical problems in the theory of the Riemann zeta-function. The study of the zeros of Hardy function's derivatives $Z^{(j)}(t)$ is the generalization of such problem. Let $T>0$. Let us define the quantity $H_j(T)$, the distance from $T$ to the nearest real zero not less than $T$ of the $j$-th derivative of the Hardy function. In the paper, an upper bound for $H_j(T)$ is proved.

Détail

Keywords: Hardy function, Riemann zeta function, exponential pair, trigonometric sum, critical line, odd order zero.