An approximation of the Hurwitz zeta function by a finite sum

Abstract

We obtain the following version of the approximation of the Hurwitz zeta-function. Let σ ≥ 0 and |t| ≤ π x. Then
ζ(s, α) = ∑_{0 ≤ n ≤ x} 1/(n + α)^s +{ (x + α)^1−s}/(s − 1) + Θ ({7√2π⁻¹ + 3}/x^σ).