The prime pairs are equidistributed among the coset lattice congruence classes

Abstract: In this paper we show that for some constant $c>0$ and for any $A>0$ there exist some $x(A)>0$ such that, If $q\leq (\log x)^{A}$ then we have \begin{align} \Psi_z(x;\mathcal{N}q(a,b),q) &= \frac{\Theta (z)}{2\phi(q)}x + O\bigg(\frac{x}{e^{c\sqrt{\log x}}}\bigg)\nonumber \end{align}for $x\geq x(A)$ for some $\Theta(z)>0$. In particular for $q\leq (\log x)^{A}$ for any $A>0$\begin{align}\Psi_z(x;\mathcal{N}_q(a,b),q)\sim \frac{x\mathcal{D}(z)}{2\phi(q)}\nonumber \end{align}for some constant $\mathcal{D}(z)>0$ and where $\phi(q)= # {(a,b):(p_i,p{i+z})\in \mathcal{N}_q(a,b)}$.