Effective estimates of exponential sums over primes

  • Hédi Daboussi
Part of the Progress in Mathematics book series (PM, volume 138)

Abstract

The asymptotic behavior of the sum
$$S(x,\alpha ) = \sum\limits_{n \leqslant x} {\pi (n)e(n\alpha ),}$$
where α is real, e(α) = e2πiα, and Λ is the von Mangoldt function, has been extensively studied by many authors. It plays a central role in Vinogradov’s solution of the 3-primes conjecture [10]. It is also a main tool in the study of the equidistribution of the sequence {, p prime} modulo 1.

Keywords

Convolution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Daboussi, On a convolution method, Congreso de teoria de los números, Universidad del Pais Vasco (1984), 110–138.Google Scholar
  2. [2]
    H. Davenport, Multiplicative Number Theory, Springer Verlag, New York 1980.MATHGoogle Scholar
  3. [3]
    F. Dress, H. Iwaniec, G. Tenenbaum Sur une somme liée á la fonction de Möbius, J. Reine Angew. Math 340 (1983), 53–58.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    H. Halberstam and H.–E. Richert, Sieve Methods, Academic Press, London 1974.MATHGoogle Scholar
  5. [5]
    A. Page, On the number of primes in an arithmetic progression, Proc. London Math. Soc. 39 (2) (1935), 116–141.CrossRefGoogle Scholar
  6. [6]
    K. Prachar, Primzahlverteilung, Springer-Verlag, Berlin 1957.MATHGoogle Scholar
  7. [7]
    C. L. Siegel, Über die Klassenzahl quadratischer Körper, Acta Arithmetica 1 (1936), 83–86.Google Scholar
  8. [8]
    G. Tenenbaum, Introduction à la théorie analytique et probabiliste desnombres, Revue de l’Institut Elie Cartan 13 (1990), Université de Nancy I.Google Scholar
  9. [9]
    R. Vaughan, Sommes trigonométriques sur les nombres premiers, C. R. Acad. Sci. Paris, Sér A 285 (1977), 981–983.MATHMathSciNetGoogle Scholar
  10. [10]
    I. M. Vinogradov, The method of trigonometrical sums in the theory of numbers, Interscience Publ., London (1954).MATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1996

Authors and Affiliations

  • Hédi Daboussi
    • 1
  1. 1.Département de MathématiquesUniversité Paris SudOrsayFrance

Personalised recommendations