# Linear inequalities in primes

## TL;DR

This paper establishes an asymptotic count for solutions to systems of linear inequalities in primes with fewer variables than previous methods, extending the Green-Tao-Ziegler theorem.

## Contribution

It improves the variable requirement from 2m+1 to m+2 for m inequalities and generalizes existing results on linear equations in primes.

## Key findings

- Proves asymptotic formula for solutions in primes
- Reduces variable count needed for such solutions
- Suggests a conjecture on sieve weights pseudorandomness

## Abstract

In this paper we prove an asymptotic formula for the number of solutions in prime numbers to systems of simultaneous linear inequalities with algebraic coefficients. For $m$ simultaneous inequalities we require at least $m+2$ variables, improving upon existing methods, which generically require at least $2m+1$ variables. Our result also generalises the theorem of Green-Tao-Ziegler on linear equations in primes. Many of the methods presented apply for arbitrary coefficients, not just for algebraic coefficients, and we formulate a conjecture concerning the pseudorandomness of sieve weights which, if resolved, would remove the algebraicity assumption entirely.