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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

Arithmetic Progression Multiplicative Function Effective Estimate Multiplicative Identity Prime Number Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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

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