Essential bounds of Dirichlet polynomials

Abstract

In this paper we have given conditions on exponential polynomials $P_{n}(s)$ of Dirichlet type to be attained the equality between each of two pairs of bounds, called essential bounds, $a_{P_{n}(s)}$, $\rho _{N}$ and $b_{P_{n}(s)}$, $\rho _{0}$ associated with $P_{n}(s)$. The reciprocal question has been also treated. The bounds $a_{P_{n}(s)}$, $b_{P_{n}(s)}$ are defined as the end-points of the minimal closed and bounded real interval $I= [ a_{P_{n}(s)},b_{P_{n}(s)} ]$ such that all the zeros of $P_{n}(s)$ are contained in the strip $I\times {\mathbb {R}}$ of the complex plane ${\mathbb {C}}$. The bounds $\rho _{N}$, $\rho _{0}$ are defined as the unique real solutions of Henry equations of $P_{n}(s)$. Some applications to the partial sums of the Riemann zeta function have been also showed.