A Fractional Newton Series for the Riemann Zeta Function

Abstract

In recent years much attention has been given to Newton series representations of a regularized Zeta function. Such representations are limited as they do not lead to a series expansion for the Zeta function that converges in the critical strip. In this paper, we define a fractional Newton series which serves as the meromorphic continuation of a classical Newton Series. We show that the Riemann Zeta function can be represented by a fractional Newton series in the critical strip. Under this representation the coefficients are given in terms of differences of the zeta function evaluated at the positive half integers instead of the usual situation of evaluating at the integers. Using the method of stationary phase, we derive an asymptotic formula for these zeta differences.