Trigonometric sums over primes III

Glyn Harman

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 3, page 727-740
  • ISSN: 1246-7405

Abstract

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New bounds are given for the exponential sum ∑ P ≤ p < 2 P e ( α p k ) were k ≥ 5 , p denotes a prime and e ( x ) = exp ( 2 π i x ) .

How to cite

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Harman, Glyn. "Trigonometric sums over primes III." Journal de théorie des nombres de Bordeaux 15.3 (2003): 727-740. <http://eudml.org/doc/252261>.

@article{Harman2003,
abstract = {New bounds are given for the exponential sum\begin\{equation*\} \sum \_\{P \le p &lt; 2P\}^\{\} e(\alpha p^k) \end\{equation*\}were $k \ge 5, p$ denotes a prime and $e(x) = \exp (2\pi ix)$.},
author = {Harman, Glyn},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {3},
pages = {727-740},
publisher = {Université Bordeaux I},
title = {Trigonometric sums over primes III},
url = {http://eudml.org/doc/252261},
volume = {15},
year = {2003},
}

TY - JOUR
AU - Harman, Glyn
TI - Trigonometric sums over primes III
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 3
SP - 727
EP - 740
AB - New bounds are given for the exponential sum\begin{equation*} \sum _{P \le p &lt; 2P}^{} e(\alpha p^k) \end{equation*}were $k \ge 5, p$ denotes a prime and $e(x) = \exp (2\pi ix)$.
LA - eng
UR - http://eudml.org/doc/252261
ER -

References

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  2. [2] E. Fouvry, P. Michel, Sur certaines sommes d'exponentielles sur les nombres premiers. Ann. Sci. Ec. Norm. Sup. IV Ser.31 (1998), 93-130. Zbl0915.11045MR1604298
  3. [3] A. Ghosh, The distribution of αp2 modulo one. Proc. London Math. Soc. (3) 42 (1981), 252-269. Zbl0447.10035
  4. [4] G. Harman, Trigonometric sums over primes I. Mathematika28 (1981), 249-254. Zbl0465.10029MR645105
  5. [5] G. Harman, Trigonometric sums over primes II. Glasgow Math. J.24 (1983), 23-37. Zbl0504.10017MR685480
  6. [6] D.R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity. Can. J. Math.34 (1982), 1365-1377. Zbl0478.10024MR678676
  7. [7] K. Kawada, T.D. Wooley, On the Waring-Goldbach problem for fourth and fifth powers. Proc. London Math. Soc. (3) 83 (2001), 1-50. Zbl1016.11046MR1829558
  8. [8] R.C. Vaughan, Mean value theorems in prime number theory. J. London Math. Soc. (2) 10 (1975), 153-162. Zbl0314.10028MR376567
  9. [9] R.C. Vaughan, The Hardy-Littlewood Method second edition. Cambridge University Press, 1997. Zbl0868.11046MR1435742
  10. [10] I.M. Vinogradov, Some theorems concerning the theory of primes. Rec. Math. Moscow N.S. 2 (1937), 179-194. Zbl0017.19803JFM63.0131.05
  11. [11] I.M. Vinogradov, A new estimation of a trigonometric sum containing primes. Bull. Acad. Sci. URSS Ser. Math.2 (1938), 1-13. Zbl0018.39002JFM64.0983.01
  12. [12] K.C. Wong, On the distribution of αpk modulo one. Glasgow Math. J.39 (1997), 121-130. Zbl0880.11052
  13. [13] T.D. Wooley, New estimates for Weyl sums. Quart. J. Math. Oxford (2) 46 (1995), 119-127. Zbl0855.11043MR1326136

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