The best known error term (f(x) in the original question) is in

E.A. Vasil’kovskaja, Mertens’ formula for an arithmetic progression, Taškent. Gos. Univ. Nauˇcn. Trudy, VoprosyMat. 548 (1977) 14–17, 139–140

See also the introduction of my paper with Zaccagnini on j.number theory, 127 (2007), 37–46, https://doi.org/10.1016/j.jnt.2006.12.015

Unconditionally, the best known error term (f(x) in the original question) is in

E.A. Vasil’kovskaja, Mertens’ formula for an arithmetic progression, Taškent. Gos. Univ. Nauˇcn. Trudy, VoprosyMat. 548 (1977) 14–17, 139–140

See also the introduction of my paper with Zaccagnini on j.number theory, 127 (2007), 37–46, https://doi.org/10.1016/j.jnt.2006.12.015