Linear inequalities in primes

  title={Linear inequalities in primes},
  author={Aled Walker},
  journal={Journal d'Analyse Math{\'e}matique},
  • A. Walker
  • Published 15 January 2019
  • Mathematics
  • Journal d'Analyse Mathématique
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… 
Gowers norms control Diophantine inequalities
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
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
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


Irrational Linear Forms in Prime Variables
Abstract We apply a recent refinement of the Hardy–Littlewood method to obtain an asymptotic lower bound for the number of solutions of a linear diophantine inequality in three prime variables. Using
Linear equations in primes
Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation
Gowers norms control diophantine inequalities
A central tool in the study of systems of linear equations with integer coefficients is the Generalised von Neumann Theorem of Green and Tao. This theorem reduces the task of counting the weighted
Roth's theorem in the primes
We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood
Asymptotic lower bounds for diophantine inequalities
§1. Introduction. In 1946, Davenport and Heilbronn [9] proved a result which opened up the study of Diophantine inequalities. Suppose that Q(x) is a diagonal quadratic form with non-zero real
The primes contain arbitrarily long arithmetic progressions
We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi�s theorem, which asserts that any subset of the integers of
A new form of the circle method, and its application to quadratic forms.
If the coefficients r(n) satisfy suitable arithmetic conditions the behaviour of F (α) will be determined by an appropriate rational approximation a/q to α, with small values of q usually producing
A higher-dimensional Siegel-Walfisz theorem
The Green-Tao-Ziegler theorem provides asymptotics for the number of prime tuples of the form $(\psi_1(n),\ldots,\psi_t(n))$ when $n$ ranges among the integer vectors of a convex body $K\subset
Abstract We prove the so-called inverse conjecture for the Gowers Us+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f : [N] → ℂ is a
Montréal notes on quadratic Fourier analysis
These are notes to accompany four lectures that I gave at the School on additive combinatorics, held in Montréal, Québec between March 30th and April 5th 2006. My aim is to introduce " quadratic