Archiv der Mathematik

, Volume 96, Issue 5, pp 417–421 | Cite as

On twisted character sums

  • Fernando ChamizoEmail author


In this paper we give a simple proof of a result by Burgess about short sums involving Dirichlet characters and exponentials. Indeed we establish a slightly stronger and more general bound that applies to sums of the form \({\sum_{n=M+1}^{M+N}f(\alpha n)\chi(n)}\), where χ is a non-principal character to the modulus p and f is a smooth 1-periodic function.

Mathematics Subject Classification (2000)

Primary 11L05 Secondary 11L40 


Character sums Burgess estimate 


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  1. 1.
    E. Bombieri, Counting points on curves over finite fields (d’après S. A. Stepanov), in: Séminaire Bourbaki, 25ème année (1972/1973), Exp. No. 430, pp. 234–241. Lecture Notes in Math. 383, Springer-Verlag, Berlin, 1974.Google Scholar
  2. 2.
    Burgess D.A.: On character sums and L-series, II. Proc. London Math. Soc. (3) 13, 524–536 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Burgess D.A.: Mean values of character sums. Mathematika 33, 1–5 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Burgess D.A.: Partial Gaussian sums. Bull. London Math. Soc. 20, 589–592 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    H. Davenport, Multiplicative number theory, Graduate Texts in Mathematics, 74, Springer-Verlag, New York, 3rd ed., 2000. Revised and with a preface by Hugh L. Montgomery.Google Scholar
  6. 6.
    J. Friedlander, Primes in arithmetic progressions and related topics, in: Analytic number theory and Diophantine problems (Stillwater, OK, 1984), Progr. Math. 70, pp. 125–134. Birkhäuser Boston, Boston, MA, 1987.Google Scholar
  7. 7.
    H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications 53, American Mathematical Society, Providence, RI, 2004.Google Scholar
  8. 8.
    W. Schmidt, Equations over finite fields: an elementary approach, Kendrick Press, Heber City, UT, 2nd ed., 2004.Google Scholar
  9. 9.
    S. A. Stepanov, Arithmetic of algebraic curves, Monographs in Contemporary Mathematics, Consultants Bureau, New York, 1994.Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Facultad de CienciasUniversidad Autónoma de MadridMadridSpain

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