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 $q\le (\log x{)}^A$ then we have \begin{align} Ψ_z(x;\mathcal{N}_q(a,b),q) &= \frac{Θ(z)}{2ϕ(q)}x + O\bigg(\frac{x}{e^{c\sqrt{\log x}}}\bigg)\nonumber \end{align}for $x\ge x(A)$ for some $\Theta (z)>0$. In particular for $q\le (\log x{)}^A$ for any $A>0$\begin{align}Ψ_z(x;\mathcal{N}_q(a,b),q)\sim \frac{x\mathcal{D}(z)}{2ϕ(q)}\nonumber \end{align}for some constant $\mathcal{D}(z)>0$ and where $\varphi (q)=\#\{(a,b):(p_i,p_{i+z})\in {\mathcal{N}}_q(a,b)\}$.

arXiv comment15 pages; submitted to Journal. arXiv admin note: text overlap with arXiv:1707.03265