Most of the proofs of Dirichlet's theorem on primes in arithmetic progressions actually give a Mertens-like theorem, and then the (weaker) statement

Chebyshev-like bound : if $(a,q) = 1$ then

$$\sum_{\substack{n \leq X \\ n \equiv a \mod q}} \Lambda(n) \gg_q \frac{X}{\varphi(q)} $$ (the factor $\varphi (q)$ is introduced here only for cosmetic reasons)

There are basically two ways in which this could be strenghtened :

• for fixed $q$, in the $X$-aspect : this amounts to replace the $\gg $ above by $\sim$, which is exactly the prime number theorem in arithmetic progressions.
• in the $q$-aspect : one asks for explicit $\epsilon$ (depending on $q$) satisfying $$\sum_{\substack{n \leq X \\ n \equiv a \mod q}} \Lambda(n) \geq \epsilon \frac{X}{\varphi(q)} $$

If complex analysis is allowed, the Siegel-Walfisz solves both problems and gives $\epsilon = 1 - o(1)$ in the range $q \ll \left( \log X \right) ^A $ (for any $A>0$). But I'm especially interested in elementary methods (with the usual meaning of the word "elementary" in this context). Following step by step Dirichlet's proof (or at least one of its modern variants), I managed to prove that $$ \epsilon = e^{- C \varphi(q) \left( \log q \right)^9} $$ is admissible. Apart from the unimportant $\log$ factors, I haven't improved this yet. Hence my questions :

What is the best (known) lower bound on $\epsilon$ that one can reach by elementary methods ?

What is the wider allowed range for $q$ that one gets from the elementary proofs of the prime number in arithmetic progressions ?

References are welcome, I've found none so far.