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 Chebyshevskii Sb., 2019, Volume 20, Issue 4, Pages 306–329 (Mi cheb850)

Zeros of the Davenport–Heilbronn function in short intervals of the critical line

Z. Kh. Rakhmonova, Sh. A. Khayrulloevb, A. S. Aminova

a Dzhuraev Institute of Mathematics, Academy of Sciences of Republic of Tajikistan, Dushanbe
b Tajik National University, Dushanbe

Abstract: Davenport and Heilbronn introduced the function $f(s)$ and showed that $f(s)$ satisfies the Riemannian type functional equation, however, the Riemann hypothesis fails for $f(s)$, and moreover, the number of zeros of $f(s)$ in the region $Re s>1$, $0<Im s le T$ exceeds $cT$, where $c>0$ is an absolute constant. S.M. Voronin proved that, nevertheless, the critical line $Re s=\frac12$ is an exceptional set for the zeros of $f(s)$, i.e. for $N_0(T)$, where $N_0(T)$ is the number of zeros of $f(s)$ on the interval $Re s=\frac12$, $0<Im s\le T$, we have the estimate $N_0(T)>cT\exp(0.05\sqrt{\ln\ln\ln\ln T})$, where $c>0$ is an absolute constant, $T\ge T_0>0$. While studying the number of zeros of the function $f(s)$ in short intervals of the critical line, A.A. Karatsuba, proved: if $\varepsilon$ and $\varepsilon_1$ are arbitrarily small fixed positive numbers not exceeding $0.001$; $T\geq T_0(\varepsilon,\varepsilon_1)>0$ and $H=T^{\frac{27}{82}+\varepsilon_1}$, then we have
$$N_0(T+H)-N_0(T)\ge H(\ln T)^{\frac{1}{2}-\varepsilon}.$$
This paper demonstrates that for the number of zeros of the Davenport-Heilbronn function $f(s)$ in short intervals of the form $[T,T+H]$ of the critical line the last relationship holds for $H\ge T^{\frac{131}{416}+\varepsilon_1}$. In particular, this result is an application of a new, in terms of exponential pairs, estimates of special exponential sums $W_j(T)$, $j=0,1,2$ which are uniform across parameters, where the problem of the non-triviality of estimates for these sums with respect to the parameter $H$ is reduced to the problem of finding the exponential pairs..

Keywords: Davenport-Heilbronn function, exponential pair, Riemann hypothesis, Selberg soothing factors.

DOI: https://doi.org/10.22405/2226-8383-2018-20-4-306-329

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UDC: 511.32
Accepted:20.12.2019

Citation: Z. Kh. Rakhmonov, Sh. A. Khayrulloev, A. S. Aminov, “Zeros of the Davenport–Heilbronn function in short intervals of the critical line”, Chebyshevskii Sb., 20:4 (2019), 306–329

Citation in format AMSBIB
\Bibitem{RakKhaAmi19} \by Z.~Kh.~Rakhmonov, Sh.~A.~Khayrulloev, A.~S.~Aminov \paper Zeros of the Davenport--Heilbronn function in short intervals of the critical line \jour Chebyshevskii Sb. \yr 2019 \vol 20 \issue 4 \pages 306--329 \mathnet{http://mi.mathnet.ru/cheb850} \crossref{https://doi.org/10.22405/2226-8383-2018-20-4-306-329}