Prime numbers in arithmetic progressions : uniformity with respect to the modulus

There is some very nice recent work of Dimitris Koukoulopoulos who uses "pretentious" methods to prove the Siegel-Walfisz Theorem. A preprint can be found here:

http://www.crm.umontreal.ca/~koukoulo/documents/publications/multfncs.pdf

Have a look at equation (5) in

Sur un théorème de Mertens in Manuscripta Mathematica 108 (2002), pages 495--513.

by myself (O. Ramaré). You can find the paper on my web site. I get $\epsilon_q = \exp(-c\sqrt{q}(\log q)^2)$ in an elementary manner. Better would improve on the Siegel zero location. Sorry, but the paper is in french :)

I hope that helps! Olivier