Estimate for growth and decay of functions in Macintyre–Evgrafov kind theorems

In the paper we obtain two results on the behavior of Dirichlet series on a real axis.
The first of them concerns the lower bound for the sum of the Dirichlet series on the system of segments $[\alpha,\,\alpha+\delta]$. Here the parameters $\alpha > 0$, $\delta > 0$ are such that $\alpha \uparrow + \infty$, $\delta \downarrow 0$. The needed asymptotic estimates is established by means of a method based on some inequalities for extremal functions in the appropriate non-quasi-analytic Carleman class. This approach turns out to be more effective than the known traditional ways for obtaining similar estimates.
The second result specifies essentially the known theorem by M. A. Evgrafov on existence of a bounded on $\mathbb{R}$ Dirichlet series. According to Macintyre, the sum of this series tends to zero on $\mathbb{R}$. We prove a spectific estimate for the decay rate of the function in an Macintyre–Evgrafov type example.