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

Z. Kh. Rakhmonov ; Sh. A. Khayrulloev ; A. S. Aminov

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 306-329

Résumé

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 $Re s>1$, $0$ 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 $Re s=\frac12$, $0$, we have the estimate $N_0(T)>cT\exp\left(0.05\sqrt{\ln\ln\ln\ln T}\right)$, 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..

Détail

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