Linear inequalities in primes

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 lea...

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Bibliographic Details
Published inJournal d'analyse mathématique (Jerusalem) Vol. 145; no. 1; pp. 29 - 127
Main Author Walker, Aled
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.12.2021
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Algebra
Analysis
Coefficients
Dynamical Systems and Ergodic Theory
Functional Analysis
Inequalities
Linear equations
Linear functions
Mathematical analysis
Mathematics
Mathematics and Statistics
Partial Differential Equations
Prime numbers
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Summary: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 2 m + 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.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-021-0174-3