Mertens-like sum in arithmetic progressions

Question

I find myself needing a good estimate for $\sum_{p\le x,\, p\equiv a\bmod q} 1/p$, perhaps something like $$ \sum_{p\le x,\, p\equiv a\bmod q} \frac1p = \frac{\log\log x}{\phi(q)} + b(q,a) + O\big(\exp(-c\sqrt{\log x})\big) $$ for $q$ up to a power of $\log x$, or something of that shape. Moreover, I'd like to be able to apply this uniformly in $q$, so the dependence of the error term on $q$ needs to be explicit (and the constant $b(q,a)$ needs to be explicit enough to manage as well).

Does anyone have a good reference to recommend for such sums?

Answer 1

This paper I believe might give such estimates:

Languasco, A.; Zaccagnini, A. A note on Mertens' formula for arithmetic progressions. J. Number Theory 127 (2007), no. 1, 37–46

Theorem 2 there works uniformly for $q\leq (\exp((\log x)^{2/5}(\log\log x)^{1/5}))^A$ for every $A>0$. Theorem 4 there works for $q\leq x$ assuming GRH. You can also get explicit $b(q,a)$.

Answer 2

Lemma 3 in "The size of L(1, χ) for real nonprincipal residue characters χ with prime modulus"

Answer 3

This may not be relevant to you anymore, but here is a precise statement on p.5.

https://arxiv.org/pdf/1309.7482.pdf

It seems that it was proved by Mertens himself in his original paper.