The prime pairs are equidistributed among the coset lattice congruence classes

TL;DR

This paper proves that prime pairs are evenly distributed among coset lattice classes under certain logarithmic bounds, providing asymptotic formulas and error estimates for their distribution.

Contribution

It establishes the equidistribution of prime pairs among coset classes with explicit asymptotic formulas and error bounds under logarithmic constraints.

Findings

Prime pairs are equidistributed among coset classes.

Asymptotic formulas for prime pair distribution with error terms.

Distribution results hold for q up to logarithmic powers.

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 $\phi(q)= \# \{(a,b):(p_i,p_{i+z})\in \mathcal{N}_q(a,b)\}$.