The Neutrix Limit of the Hurwitz Zeta Function and Its Application

In this paper, the neutrix limit is used to extend the definition of the Hurwitz zeta function ζ(α, x) and its partial derivatives to the whole complex plane except for non-positive integers α, in particular, the values of ζ(1, x) is obtained. This definition is equivalent to the Hermite’s integral of ζ(α, x) as α 6= 1, 0,−1, . . .. Moreover, some properties of ζ(1, x) are established and we find that ζ(1, x) is the inverse number of the digamma function. In addition, we pay our special attention to the closed forms of the certain integrals involving the Hurwitz zeta function, which can be expressed as a linear combination of the Riemann zeta functions and their derivatives.