On real zeros of the derivative of the Hardy function
Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 234-240
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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.
Keywords:
Hardy function, Riemann zeta function, exponential pair, trigonometric sum, critical line, odd order zero.
@article{CHEB_2021_22_5_a14,
author = {Sh. A. Khayrulloev},
title = {On real zeros of the derivative of the {Hardy} function},
journal = {\v{C}eby\v{s}evskij sbornik},
pages = {234--240},
publisher = {mathdoc},
volume = {22},
number = {5},
year = {2021},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/CHEB_2021_22_5_a14/}
}
Sh. A. Khayrulloev. On real zeros of the derivative of the Hardy function. Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 234-240. http://geodesic.mathdoc.fr/item/CHEB_2021_22_5_a14/