# Linear inequalities in primes

@article{Walker2021LinearII, title={Linear inequalities in primes}, author={Aled Walker}, journal={Journal d'Analyse Math{\'e}matique}, year={2021} }

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 require at least $2m+1$ variables. Our result also generalises the theorem of Green and Tao on linear equations in primes. Many of the methods presented apply for arbitrary coefficients, not just for algebraic coefficients…

## 3 Citations

Gowers norms control Diophantine inequalities

- Mathematics
- 2017

The classical 'Generalised von Neumann Theorem' is a central tool in the study of systems of linear equations with integer coefficients. This theorem reduces the task of counting weighted solutions…

Quantitative bounds for Gowers uniformity of the M\"obius and von Mangoldt functions

- Mathematics
- 2021

We establish quantitative bounds on the U[N ] Gowers norms of the Möbius function μ and the von Mangoldt function Λ for all k, with error terms of shapeO((log logN)−c). As a consequence, we obtain…

Correlations of sieve weights and distributions of zeros

- Mathematics
- 2021

In this note we give two small results concerning the correlations of the Selberg sieve weights. We then use these estimates to derive a new (conditional) lower bound on the variance of the primes in…

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