Distribution of the zeros of linear combinations of $L$-Dirichlet functions lying on the critical line

Some problems of the number theory are associated with the zeros of special functions, such as the Riemann zeta function $\zeta(s)$, Dirichlet $L$-functions $L(s,\chi)$ and others. The Riemann zeta function is the most famous. On the half-plane $\Re{s}>1$, the Riemann zeta function is defined by Dirichlet series
$$\zeta(s)=\sum_{n=1}^{+\infty}n^{-s}.$$

In 1859, Riemann conjectured that all non-trivial zeros of the Riemann zeta function lie on the critical line $\Re{s}=\frac{1}{2}$. Hardy was the first to prove in 1914 that an infinity of zeros are on the critical line. In 1942, Selberg obtained lower bound of the correct order of magnitude for the number zeros of the Riemann zeta functions on intervals of critical line $[T,T+H], H=T^{0.5+\varepsilon}$. In 1984, A. A. Karatsuba proved Selberg's result for shorter intervals of critical line $[T,T+H], H=T^{\frac{27}{82}+\varepsilon}$. For arithmetic Dirichlet series satisfying a functional equation of Riemann type but admitting no Euler product expansions, lower bounds of the correct order of magnitude for the number of their zeros on intervals of the critical line $\Re s =1/2$ have not been obtained so far. The first to show that the critical line contains abnormally many zeros of an arithmetic Dirichlet series without Euler products was Voronin, who proved in 1980 that interval $(0,T]$ of critical line contains more than
$$ cTe^{\frac{1}{20}\sqrt{\ln{\ln{\ln{\ln{T}}}}}}$$
zeros of the Davenport–Heilbronn function. In 1989 A. A. Karatsuba developed a new method for obtaining lower bounds for the number zeros of certain Dirichlet series in intervals of critical line; by using this method, he substantially strengthened Voronin’s result. In 1991 Karatsuba solved (by his 1989 method) the problem of estimating the number zeros of linear combinations of functions which are analogous the Hardy function.
In the present paper we prove a theorem similar to the theorem of A. A. Karatsuba (in 1991), but only for "almost all" intervals of the form $(T, T + H)$, $H = X^{\varepsilon}$, where $\varepsilon$ is an arbitrary positive number, and $X \leq T \leq 2X$, $X>X_0(\varepsilon)$.
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