Skip to main content Accessibility help


  • Roger Baker (a1)


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].



Hide All
1. Baker, R. C., Diophantine Inequalities (London Mathematical Society Monographs New Series 1 ), Oxford University Press (Oxford, 1986).
2. Baker, R. C., Correction to “Weyl sums and Diophantine approximation [J. Lond. Math. Soc. (2) 25 (1982), 25–34]”. J. Lond. Math. Soc. (2) 46 1992, 202204.
3. Baker, R. C., Small fractional parts of polynomials. Funct. Approx. Comment. Math. 55 2016, 131137.
4. Baker, R. C. and Harman, G., On the distribution of 𝛼p k modulo one. Mathematika 48 1991, 170184.
5. Baker, R. C. and Weingartner, A., A ternary Diophantine inequality over primes. Acta Arith. 162 2014, 159196.
6. Bourgain, J., Demeter, C. and Guth, L., Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. of Math. (2) 184 2016, 633682.
7. de Bruijn, N. G., On the number of positive integers ⩽x and free of prime factors > y. II. Nederl. Akad. Wetensch. Proc. Ser. A 69 1966, 239247.
8. Cochrane, T., Exponential sums modulo prime powers. Acta Arith. 101 2002, 131149.
9. Harman, G., Trigonometric sums over primes I. Mathematika 28 1981, 249254.
10. Harman, G., Trigonometric sums over primes II. Glasg. Math. J. 24 1983, 2337.
11. Harman, G., On the distribution of 𝛼p modulo one. II. Proc. Lond. Math. Soc. (3) 72 1996, 241260.
12. Harman, G., Prime-detecting Sieves, Princeton University Press (Princeton, NJ, 2007).
13. Hooley, C., On an elementary inequality in the theory of Diophantine approximation. In Analytic Number Theory (Allerton Park, IL, 1995), Vol. 2 (Progress in Mathematics 139 ), Birkhäuser (Boston, MA, 1996), 471486.
14. Matomaki, K., The distribution of 𝛼p modulo one. Math. Proc. Cambridge Philos. Soc. 147 2009, 267283.
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.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed