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FRACTIONAL PARTS OF POLYNOMIALS OVER THE PRIMES

Part of: Diophantine approximation, transcendental number theory Multiplicative number theory Exponential sums and character sums

Published online by Cambridge University Press:  29 November 2017

Roger Baker
Roger Baker*
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A. email baker@math.byu.edu
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Abstract

Let $f$ be a polynomial of degree $k>1$ with irrational leading coefficient. We obtain results of the form

$$\begin{eqnarray}\Vert f(p)\Vert <p^{-\unicode[STIX]{x1D70E}}\end{eqnarray}$$
for infinitely many primes $p$ that supersede those of Harman [Trigonometric sums over primes I. Mathematika28 (1981), 249–254; Trigonometric sums over primes II. Glasg. Math. J.24 (1983), 23–37] and Wong [On the distribution of $\unicode[STIX]{x1D6FC}p^{k}$ modulo 1. Glasg. Math. J.39 (1997), 121–130].

MSC classification

Primary: 11J54: Small fractional parts of polynomials and generalizations
Secondary: 11L20: Sums over primes 11N36: Applications of sieve methods
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 715 - 733
Copyright
Copyright © University College London 2017 

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