Uniform asymptotic approximations for incomplete Riemann Zeta functions

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 t + 1 ) - 1 d t 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 Re α > 0 and α ≠ 1 . The asymptotic behaviour of Z 1 ( α , x ) and Z 2 ( α , x ) is studied for the case Re α = σ > 0 fixed and Im α = τ → ∞ , and using Liouville–Green (WKBJ) analysis, asymptotic approximations are obtained, complete with explicit error bounds, which are uniformly valid for 0 ⩽ x < ∞ .