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FRACTIONAL PARTS OF POLYNOMIALS OVER THE PRIMES

  • Roger Baker (a1)

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. Mathematika 28 (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].

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References

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12. Harman, G., Prime-detecting Sieves, Princeton University Press (Princeton, NJ, 2007).
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15. Vaughan, R. C., The Hardy–Littlewood Method (Cambridge Tracts in Mathematics 125 ), 2nd edn., Cambridge University Press (Cambridge, 1997).
16. Wong, K. C., On the distribution of 𝛼p k modulo 1. Glasg. Math. J. 39 1997, 121130.
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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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