FRACTIONAL PARTS OF POLYNOMIALS OVER THE PRIMES. II. Issue 3 (26th June 2018)

• Record Type: Journal Article
• Authors: Baker, Roger
• Is Part Of: Mathematika. Volume 64:Issue 3(2018)
• Page Start: 742
• Page End: 769
• Publication Date: 2018-06-26
• Subjects: 10J54 (primary) -- 11L20 -- 11N36 (secondary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579318000244
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract : Let | ⋯ | denote distance from the integers. Let α, β, γ be real numbers with α irrational. We show that the inequality ‖ α p 2 + β p + γ ‖ < p − 3 / 17 + ε has infinitely many solutions in primes p, sharpening a result due to Harman [On the distribution of α p modulo one II. Proc. Lond. Math. Soc. (3) 72 (1996), 241–260] in the case β = 0 and Baker [Fractional parts of polynomials over the primes. Mathematika 63 (2017), 715–733] in the general case.