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Uniform asymptotic approximations for incomplete Riemann zeta functions. (English) Zbl 1091.33016
An incomplete Riemann zeta-function 𝒵 1 (α,x) is examined, along with a complementary incomplete Riemann zeta-function 𝒵 2 (α,x). These functions are defined by 𝒵 1 (α,x)={(1-2 1-α )Γ(α)} -1 0 x t α-1 (e ' +1) -1 dt and 𝒵 2 (α,x)=ζ(α)-𝒵 1 (α,x), where ζ(α) is the classical Riemann zeta function. 𝒵 1 (α,x) has the property that lim x 𝒵 1 (α,x)=ζ(α) for Reα>0 and α1. The asymptotic behaviour of 𝒵 1 (α,x) and 𝒵 2 (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 0x<.
33E20Functions defined by series and integrals
11M06ζ(s) and L(s,χ)
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)