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At order $\alpha_s^4$ in perturbative quantum chromodynamics, even-integer$\zeta$-function values are present in Euclidean physical correlation functionslike the scalar quark correlation function or the scalar gluonium correlator.We demonstrate that these contributions cancel when the perturbative expansionis expressed in terms of the so-called $C$-scheme coupling $\hat\alpha_s$ whichhas recently been introduced in Ref. [1]. It is furthermore conjectured that a$\zeta_4$ term should arise in the Adler function at order $\alpha_s^5$ in the$\overline{\rm MS}$-scheme, and that this term is expected to disappear in the$C$-scheme as well.

Absence of even-integer $\zeta$-function values in Euclidean physical quantities in QCD