On the sharpness of the Hayman-type analogue of the Wiman inequality for entire Dirichlet series

Abstract

We prove the sharpness of Sheremeta's analog of the Wiman-type inequality for entire Dirichlet series.  Consider $ F(z) = \sum_{n=0}^{+\infty} a_n e^{z \lambda_n},\ $ $0 = \lambda_0 < \lambda_n \uparrow +\infty\ (1 \le n \to +\infty)$ and $ |n(t) - \Delta t^\rho| \le \mathcal{D}\ (t \ge t_0),\ \Delta\in\mathbb{R}_+,\ \mathcal{D}\in\mathbb{R}_+,\ \rho \ge \frac{1}{2},\ n(t) = \sum_{n\colon  \lambda_n \le t} 1.  $ Then there exists a set $E$ of finite measure and for each $k \in \mathbb{N}$ there exists a number $\sigma_k$ such that for all $\sigma \in [\sigma_k; +\infty) \backslash E$ we have \begin{equation*} M(\sigma, F) \le \mu(\sigma, F)\cdot \ln^{\rho - \frac{1}{2}} \mu(\sigma, F)\ln_2^{\rho} \mu(\sigma, F)\cdots\ln_k^{\rho} \mu(\sigma, F)\cdot \ln_{k+1}^{\rho+\delta} \mu(\sigma, F).  \end{equation*}   In obtained inequality all powers of each iteration of the logarithm of maximal term are sharp. Also are obtained analogues of this result for entire gap power series.