On the fractional moments of some mollified arithmetical Dirichlet series

In 2002, A.A. Karatsuba demonstrated that the true order of fractional moments of some Dirichlet series allows one to obtain the estimate for the number of zeta-function zeros on the critical line which is more precise than the estimate of G. Hardy and J. Littlewood (1921).
In 2017, the author has found the true order lower and upper estimates for some mollified $L$ -functions and has applied it to the estimate of the number of zeros of Davenport-Heilbronn function on the critical line. The moments of mollified Dirichlet $L$ -functions mean the integrals
$$ \int_{T}^{2T}\bigl|L\bigl(\tfrac{1}{2}+it,\chi\bigr)\phi\bigl(\tfrac{1}{2}+it\bigr)\bigr|^{2k}dt, $$
where the function $\phi\bigl(\tfrac{1}{2}+it\bigr)$ has no odd zeros and such that the function $L\bigl(\tfrac{1}{2}+it,\chi\bigr)$ is such close to the constant function as possible. The idea of introducing such mollifier function $\phi\bigl(\tfrac{1}{2}+it\bigr)$ belongs to A. Selberg.
The exponent $2k$ is called the order of the moment. In his previous studies, the author considered only the moments of orders $\tfrac{1}{2}$ and $1$. In the present talk, we present the estimates of moments of orders $\tfrac{2}{v}$, where $v$ is any integer greater that $2$.
Our main result is the following. Suppose that $\varepsilon$ is an arbitrary small number, $X=T^{\,\varepsilon}$, and let
$$ \sum_{\nu=1}^{\infty}\frac{\alpha(\nu)}{\nu^{s}}\,=\, \prod\limits_{p\equiv\pm 1(\mmod 5)}\biggl(1-\frac{1}{2vp^{s}}\biggr)\!\!\prod\limits_{p\equiv\pm 2(\mmod 5)}\biggl(1-\frac{\varepsilon}{p^{s}}\biggr), $$

\begin{equation*} \beta(\nu)\,=\, \begin{cases} \displaystyle \alpha(\nu)\chi_{1}(\nu)\biggl(1-\frac{\log{\nu}}{\log X\mathstrut }\biggr), & \text{if}\;\;\nu<X,\\ 0, & \text{if}\;\;\nu\ge X, \end{cases} \end{equation*}
where $\chi_{1}(\nu)$ is Dirichlet character modulo $5$ with the condition $\chi_{1}(2)=i$,
$$ \varphi\bigl(\tfrac{1}{2}+it\bigr)\,=\,\sum_{\nu<X}\frac{\beta(\nu)}{\nu^{\,1/2+it}},\quad \phi\bigl(\tfrac{1}{2}+it\bigr)=\bigl(\varphi\bigr(\tfrac{1}{2}+it\bigr)\bigr)^{2v}. $$

theorem. The following estimates hold true:
\begin{multline*} T(\log T)^{(1+2\varepsilon v)^{2}/(2v^2)}\ll \int_T^{2T}\bigl|L\bigl(\tfrac{1}{2}+it,\overline{\chi}_{1}\bigr)\phi\bigl(\tfrac{1}{2}+it\bigr)\bigr|^{2/v}dt\ll\\ \ll T(\log T)^{(1+2\varepsilon v)^{2}/(2v^2)},\\ \int_T^{2T}\bigr|L\bigl(\tfrac{1}{2}+it,\chi_{1}\bigr)\phi\bigl(\tfrac{1}{2}+it\bigr)\bigr|^{2/v}dt\ll T(\log T)^{(1-2\varepsilon v)^{2}/(2v^2)},\\ \int_T^{2T}\bigr|L\bigl(\tfrac{1}{2}+it,\overline{\chi}_{1}\bigr)\phi\bigl(\tfrac{1}{2}+it\bigr)\bigr|dt\ll T(\log T)^{(1+2\varepsilon v)^{2}/8},\\ \int_T^{2T}\bigl|L\bigl(\tfrac{1}{2}+it,\chi_{1}\bigr)\phi\bigl(\tfrac{1}{2}+it\bigr)\bigr|dt\ll T(\log T)^{(1-2\varepsilon v)^{2}/8}. \end{multline*}

Denote by $N_{0}(T)$ the number of zeros of Davenport-Heilbronn function on the segment $\bigl[\tfrac{1}{2},\tfrac{1}{2}+iT\bigr]$. Then the above theorem implies that
$$ N_{0}(2T)\,-\,N_{0}(T)\,\gg\,T(\log T)^{1/2+1/12-\varepsilon}. $$