Uniform asymptotic approximations for incomplete Riemann Zeta functions

An incomplete Riemann Zeta function Z1(α,x)Z1(α,x) is examined, along with a complementary incomplete Riemann Zeta function Z2(α,x)Z2(α,x). These functions are defined by Z1(α,x)={(1-21-α)Γ(α)}-1∫0xtα-1(et+1)-1dt and Z2(α,x)=ζ(α)-Z1(α,x)Z2(α,x)=ζ(α)-Z1(α,x), where ζ(α)ζ(α) is the classical Riemann Zeta function. Z1(α,x)Z1(α,x) has the property that limx→∞Z1(α,x)=ζ(α) for Reα>0 and α≠1α≠1. The asymptotic behaviour of Z1(α,x)Z1(α,x) and Z2(α,x)Z2(α,x) is studied for the case Reα=σ>0 fixed and Imα=τ→∞, and using Liouville–Green (WKBJ) analysis, asymptotic approximations are obtained, complete with explicit error bounds, which are uniformly valid for 0⩽x<∞0⩽x<∞.