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.