OK, so the paper in Comment. Math. Univ. ST. Pauli was indeed a spot on reference (see answer and comments below), in turn leading to a paper H. G. Kopetzky, Über die Größenordnung der Teilerfunktion in Restklassen, Monatshefte für Mathematik, 82(4), 1976 for the values of the constants. Thank you Dietrich!

Thank you for the answer, but this is indeed, as Matt noted, the coprime case