On differences of zeta values

Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri–Lagarias, Maślanka, Coffey, Báez-Duarte, Voros and others. We apply the theory of Nörlund–Rice integrals in conjunction with the saddle-point method and derive precise asymptotic estimates. The method extends to Dirichlet L-functions and our estimates appear to be partly related to earlier investigations surrounding Liʹs criterion for the Riemann hypothesis.