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FRACTIONAL PARTS OF POLYNOMIALS OVER THE PRIMES. II

Part of: Multiplicative number theory Exponential sums and character sums

Published online by Cambridge University Press:  26 June 2018

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 $\Vert \cdots \Vert$ denote distance from the integers. Let $\unicode[STIX]{x1D6FC}$, $\unicode[STIX]{x1D6FD}$, $\unicode[STIX]{x1D6FE}$ be real numbers with $\unicode[STIX]{x1D6FC}$ irrational. We show that the inequality

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FC}p^{2}+\unicode[STIX]{x1D6FD}p+\unicode[STIX]{x1D6FE}\Vert <p^{-3/17+\unicode[STIX]{x1D700}}\end{eqnarray}$$
has infinitely many solutions in primes $p$, sharpening a result due to Harman [On the distribution of $\unicode[STIX]{x1D6FC}p$ modulo one II. Proc. Lond. Math. Soc. (3)72 (1996), 241–260] in the case $\unicode[STIX]{x1D6FD}=0$ and Baker [Fractional parts of polynomials over the primes. Mathematika63 (2017), 715–733] in the general case.

MSC classification

Secondary: 11L20: Sums over primes 11N36: Applications of sieve methods
Type
Research Article
Information
Mathematika , Volume 64 , Issue 3 , 2018 , pp. 742 - 769
Copyright
Copyright © University College London 2018 

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Footnotes

Research supported in part by Collaboration Grant 412557 from the Simons Foundation.

References

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