Essential bounds of Dirichlet polynomials

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, aP(n)(s), rho(N) and bP(n)(s), rho(0) associated with P-n(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 = [aP(n)(s), bP(n)(s)] such that all the zeros of P-n(s) are contained in the strip I x R of the complex plane 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.