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.


**EDIT** : In view of the comments and answers below, I conclude that what I'm asking for is not as classic I thought it was. I summarize the state of the question :


* there are (relatively easy and) elementary proofs of Siegel's theorem, but deducing from it a Siegel-Walfisz theorem seems to require complex (or Fourier) analysis.
* No elementary proof of Linnik's theorem exists *in the literature*, but Micah Milinovich suggests below that A. Granville could have further information on this subject. It may be worth contacting him.


**EDIT 2** : My questions are essentially solved by the "anonymous's" comment below. There indeed exists an elementary proof of Linnik's theorem (or at least a proof avoiding most of the complex analysis machinery in original Linnik's proof). The last use of complex analysis originated results lies in the explicit use of $\Psi(X,q,a) = \frac{X-\frac{X^{\beta}}{\beta}}{\varphi(q)} + \text{Error term}$ (according to Andrew Granville, this seems to be fixed, but details are not clear


Thanks for your help !



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>


