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Chebyshevskii Sbornik, 2019, Volume 20, Issue 4, Pages 371–384
DOI: https://doi.org/10.22405/2226-8383-2019-20-4-371-384
(Mi cheb854)
 

This article is cited in 1 scientific paper (total in 1 paper)

On the functions of Hardy zeros and its derivatives lying on the critical line

Sh. A. Khayrulloev

Tajik National University, Dushanbe
Full-text PDF (659 kB) Citations (1)
References:
Abstract: Hardy function $Z(t)$ takes real values for real values of $t$ and real zeros of $Z(t)$ are the zeros of $\zeta(s)$, that are on the critical line. The first result of the zeros of the Riemann zeta function on the critical line is the G.Hardy theorem. In 1914 he proved that $\zeta(1/2+it)$ has infinitely many real zeros. Then Hardy and Littlewood in 1921 proved that the interval $(T, T+H)$ when $H\ge T^{1/4+\varepsilon}$ contains odd order zero of function $\zeta(1/2+it)$. Jan Moser in 1976 proved that this assertion holds for $H\ge T^{1/6}\ln^2T$. In 1981 A.A.Karatsuba proved Hardy-Litllvud theorem already for $H\ge T^{5/32}\ln^2T$.
In 2006 Z.Kh.Rahmonov, Sh.A.Khayrulloev have reduced a problem of the magnitude of the interval $(T, T+H)$ of the critical line, which contains an odd order zero of the zeta function to the problem of finding an exponential pairs for estimating the special trigonometric sums.
In 2009 Z.H.Rahmonov, Sh.A.Khayrulloev find the lower bound of $\theta_1(k, l)$ on $\mathcal{P}$ – the set of all exponential pairs $(k, l)$ of different from $(1/2, 1/2)$ and having the form:
$$ \mathop {\inf }\limits_{(k,l) \in \mathcal{P} } \theta _1 (k;l) = R + 1, $$
where $R = 0.8290213568591335924092397772831120\ldots $ – Rankin constant.
In 1981 A.A.Karatsuba has studied the problem of neighboring zeros of the function $Z(t)$ together with the problem of the neighboring extremum points or of inflection points of the function $Z(t)$ or in a more general substitution – of the neighboring zeros of functions $Z^{(j)}(t)$, $j\ge 1$.
The main result of our work is to reduce the problem of the magnitude of the interval $(T, T+H)$ of the critical line, which is known to be a odd order zero of the function $Z^{(j)}(t)$, ($j\ge 1$) to the problem of finding the exponential pairs for estimating the special trigonometric sum and to improve the A.A.Karatsuby theorem for $j=1$.
Keywords: Hardy function, exponential pair, critical line, the Riemann zeta function.
Received: 15.11.2019
Accepted: 20.12.2019
Document Type: Article
UDC: 511.3
Language: Russian
Citation: Sh. A. Khayrulloev, “On the functions of Hardy zeros and its derivatives lying on the critical line”, Chebyshevskii Sb., 20:4 (2019), 371–384
Citation in format AMSBIB
\Bibitem{Kha19}
\by Sh.~A.~Khayrulloev
\paper On the functions of Hardy zeros and its derivatives lying on the critical line
\jour Chebyshevskii Sb.
\yr 2019
\vol 20
\issue 4
\pages 371--384
\mathnet{http://mi.mathnet.ru/cheb854}
\crossref{https://doi.org/10.22405/2226-8383-2019-20-4-371-384}
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  • https://www.mathnet.ru/eng/cheb/v20/i4/p371
  • This publication is cited in the following 1 articles:
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