On the zeroes of the function of Davenport and Heilbronn lying on the critical line

Let $\chi_1(n)$ be the character of Dirichlet $\mod 5$ such that $\chi_1(2)=i$,
$$ \varkappa\,=\,\frac{\sqrt{10-2\sqrt{5}}-2}{\sqrt{5}-1}. $$
Davenport–Heilbronn function is defined as follows:
$$ f(s)\,=\,\frac{1-i\varkappa}{2}L(s,\chi_1)\,+\,\frac{1+i\varkappa}{2}L(s,\overline{\chi}_1). $$
The function $f(s)$ was introduced and investigated by Davenport and Heilbronn in 1936. It satisfies the functional equation $g(s)=g(1-s)$ of Riemann's type where
$$ g(s)\,=\,\biggl(\frac{\pi}{5}\biggr)^{\!-\,s/2}\Gamma\biggl(\frac{1+s}{2}\biggr)f(s). $$


However, it is well -known however, that not all non -trivial zeros of $f(s)$ lie on the line $\Re s=\frac{1}{2}$.

In the region $\Re s>1$, $0<\Im s\le T$, the number of zeros of $f(s)$ exceeds $cT$, where $c>0$ is an absolute constant (Davenport and Heilbronn, 1936).

Moreover, the number of zeros of $f(s)$ in the region $\tfrac{1}{2}<\sigma_1<\Re s<\sigma_2$, $0<\Im s\le T$ exceeds $c_{1}T$, where $c_{1}>0$ is an absolute constant (S.M. Voronin, 1976).

In 1980, Voronin proved that “abnormally many” zeros of $f(s)$ lie on the critical line $\Re s=\tfrac{1}{2}$. Let $N_{0}(T)$ be the number of zeros of $f(s)$ on the segment $\Re s=\tfrac{1}{2}$, $0<\Im s\le T$. Then Voronin got the estimate
$$ N_{0}(T)\,>\,c_{2}T\exp\bigl(\tfrac{1}{20}\sqrt{\log\log\log\log T}\bigr), $$
where $c_{2}>0$ is an absolute constant.

In 1990, A.A. Karatsuba improved Voronin's estimate significantly and got the inequality
$$ N_{0}(T)\,>\,T(\log T)^{1/2-\varepsilon}, $$
where $\varepsilon>0$ is an arbitrary small constant, $T>T_{0}(\varepsilon)>0$.

In 1994, A.A. Karatsuba got somewhat more precise estimate
$$ N_{0}(T)\,>\,T(\log T)^{1/2}\exp{\bigl(-c_{3}\sqrt{\log\log T}\bigr)}, $$
where $c_{3}>0$ is an absolute constant.

In this talk, we represent the the following theorem proved by the author.

Theorem. Let $\varepsilon>0$ be an arbitrary small constant. Then the estimate
$$ N_{0}(T)\,>\,T(\log T)^{1/2+1/16-\varepsilon}. $$
holds.