Absence of even-integer ζ-function values in Euclidean physical quantities in QCD

At order αₛ⁴ in perturbative quantum chromodynamics, even-integer ζ-function values are present in Euclidean physical correlation functions like the scalar quark correlation function or the scalar gluonium correlator. We demonstrate that these contributions cancel when the perturbative expansion is expressed in terms of the so-called C-scheme coupling αₛ which has recently been introduced in Ref. [1]. It is furthermore conjectured that a ζ₄ term should arise in the Adler function at order αₛ⁵ in the MS̄-scheme, and that this term is expected to disappear in the C-scheme as well.