The prime number theorem tells us that , instead $\pi\left(x\right)$ the number of primes less than or equal to $x$, we have $$\pi\left(x\right)\sim\frac{x}{\log x}.$$ In a similar manner considered $1\leq a \leq q$ with $(a,q)=1$ and defined $\pi\left(x,a,q\right)$ the number of primes less than or equal to $x$ congruous $a\,\textrm{mod}\, q$ and $\phi\left(n\right)$ the number of minor numbers and coprime with $n$, we have $$\pi(x,a,q)\thicksim\frac{1}{\phi(q)}\frac{x}{\log x}.$$ If $q$ is "small" you have asymptotic formulas for $\pi\left(x,a,q\right)$ (see the Siegel - Walfisz theorem). For any $q$ we have the estimate $$\pi(x,a,q)\gg\frac{1}{\phi(q)}\frac{x}{\log x}.$$ I would like to know if there is an estimate of the type $$\pi(x,a,q)\ll\frac{1}{\phi(q)}\frac{x}{\log x}$$ for any $q$. I hope I was clear! Sorry for my bad english!