For my work, i need the following Conjecture: Let $N$ large number such that exist a prime number $q$ and $A>\frac{1}{2}$ such that $N^{1/2}<N^{A}\leq q1<N.$ Then $\forall a\in\left[1,\, q\right)$ $$\frac{1}{\phi\left(q\right)}\frac{N}{\log N}\ll\pi\left(N;\, q,\, a\right)\ll\frac{1}{\phi\left(q\right)}\frac{N}{\log N}.$$ Is it wrong or in part demonstrated? Thank you!
