Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Character sums over shifted smooth numbers

Author: Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 135 (2007), 2699-2705
MSC (2000): Primary 11L40, 11N25
Published electronically: May 2, 2007
MathSciNet review: 2317942
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give nontrivial bounds in various ranges for character sums of the form

$\displaystyle \sum_{n\in\mathcal S(x,y)}\chi(n +a), \qquad \gcd(a,p) = 1, $

where $ \chi$ is a nontrivial multiplicative character modulo a prime $ p$ and $ \mathcal S(x,y)$ is the set of positive integers $ n\le x$ that are divisible only by primes $ q \le y$.

References [Enhancements On Off] (What's this?)

  • 1. W. Banks, J. B. Friedlander, M. Garaev and I. E. Shparlinski, `Character sums with exponential functions over smooth numbers', Indag. Math., 17 (2006), 157-168. MR 2217571 (2007c:11090)
  • 2. A. Granville, `Smooth numbers: Computational number theory and beyond', Proc. MSRI Conf. Algorithmic Number Theory: Lattices, Number Fields, Curves, and Cryptography, Berkeley 2000, Cambridge Univ. Press, (to appear).
  • 3. A. Granville and K. Soundararajan, `Large character sums', J. Amer. Math. Soc., 14 (2001), 365-397. MR 1815216 (2002h:11074)
  • 4. A. Hildebrand and G. Tenenbaum, `Integers without large prime factors', J. de Théorie des Nombres de Bordeaux, 5 (1993), 411-484. MR 1265913 (95d:11116)
  • 5. H. Iwaniec and E. Kowalski, Analytic number theory, Amer. Math. Soc., Providence, RI, 2004. MR 2061214 (2005h:11005)
  • 6. A. A. Karatsuba, `Sums of characters with prime numbers', Izv. Akad. Nauk, Ser. Mat., 34 (1970), 299-321. MR 0271040 (42:5923)
  • 7. R. Lidl and H. Niederreiter, Finite fields, Cambridge University Press, Cambridge, 1997. MR 1429394 (97i:11115)
  • 8. Z. Kh. Rakhmonov, `On the distribution of values of Dirichlet characters and their applications', Proc. Steklov Inst. Math., 207 (1995), 263-272. MR 1401821 (97f:11068)
  • 9. G. Tenenbaum, Introduction to analytic and probabilistic number theory, University Press, Cambridge, UK, 1995. MR 1342300 (97e:11005b)
  • 10. R. C. Vaughan, `A new iterative method for Waring's problem', Acta Math., 162 (1989), 1-71. MR 0981199 (90c:11072)
  • 11. I. M. Vinogradov, `Improvement of an estimate for the sum of the values $ \chi(p+k)$'. Izv. Akad. Nauk Ser. Mat., 17 (1953), 285-290 (in Russian). MR 0061623 (15:855g)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11L40, 11N25

Retrieve articles in all journals with MSC (2000): 11L40, 11N25

Additional Information

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia

Received by editor(s): December 19, 2005
Received by editor(s) in revised form: March 13, 2006, and May 15, 2006
Published electronically: May 2, 2007
Additional Notes: During the preparation of this paper, the author was supported in part by ARC grant DP0556431.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society