Essential bounds of Dirichlet polynomials

Abstract

In this paper we have given conditions on exponential polynomials Pn(s) of Dirichlet type to be attained the equality between each of two pairs of bounds, called essential bounds, aPn (s), ρN and bPn (s), ρ0 associated with Pn(s). The reciprocal question has been also treated. The bounds aPn (s), bPn (s) are defined as the end-points of the minimal closed and bounded real interval I = [aPn (s), bPn (s)] such that all the zeros of Pn(s) are contained in the strip I × R of the complex plane C. The bounds ρN , ρ0 are defined as the unique real solutions of Henry equations of Pn(s). Some applications to the partial sums of the Riemann zeta function have been also showed.