Oscillations of Quadratic L-Functions

  • R. C. Baker
  • Hugh L. Montgomery
Chapter
Part of the Progress in Mathematics book series (PM, volume 85)

Abstract

All real non-principal characters are of the form \( XD\left( n \right) = \left( {\frac{D}{n}} \right) \) where D belongs to the set Q of quadratic discriminants, Q={D : D is not a square and D ≡ 0 or 1 (mod 4)}.

Keywords

Assure Harman 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R.C. Baker and G. Harman, Unbalanced quadratic residues and non-residues, Proc. Camb. Philos. Soc. 98 (1975), 9–17.MathSciNetCrossRefGoogle Scholar
  2. [2]
    P.T. Bateman, G.B. Purdy and S. Wagstaff, Some numerical results on Fekete polynomials, Math. Comp. 29 (1975), 7–23.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    S.D. Chowla, Note on Dirichlet’s L-fucnctions, Acta Arith. 1 (1936), 113–114.MathSciNetGoogle Scholar
  4. [4]
    K.L Chung, A course in probability theory, Harcourt, Brace and World (New York ), 1968.MATHGoogle Scholar
  5. [5]
    E. Cohen and R.L. Robinson, On the distribution of the k-free integers in residue classes, Acta Arith. 8 (1962/3), 283–293.MathSciNetGoogle Scholar
  6. [6]
    H. Davenport, Multiplicative Number Theory, Second Edition, Springer-Verlag (New York), 1980, 178 pp.MATHGoogle Scholar
  7. [7]
    P.D.T.A. Elliott, On the mean value of f(p), Proc. London Math. Soc. (3) 21 (1970), 28–96.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    P.D.T.A. Elliott, On the distribution of the values of quadratic L-series in the half-plane σ 1/2, Invent. Math. 21 (1973), 319–338.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    H. Heilbronn, On real characters, Acta Arith. 2 (1937), 212–213.MATHGoogle Scholar
  10. [10]
    M. Jutila, On character sums and class numbers, J. Number Theory 5 (1973), 203–214.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    M. Jutila, A density estimate for L-functions with a real character, Ann. Acad. Sci. Fenn. Ser. AI 508 (1972), 10 pp.MathSciNetGoogle Scholar
  12. [12]
    M. Jutila, On the mean values of L-functions and short character sums with real characters, Acta Arith. 26 (1975), 405–410.MathSciNetMATHGoogle Scholar
  13. [13]
    M. Jutila, On the mean values of Dirichlet polynomials with real characters, Acta Arith. 27 (1975), 191–198.MathSciNetMATHGoogle Scholar
  14. [14]
    J. Kaczorowski, Some problems concerning roots of polynomials with Dirichlet characters as coefficients, in Elementary and Analytic Theory of Numbers, Henryk Iwaniec, ed., Banach Center Publ., 1985, pp. 333–337.Google Scholar
  15. [15]
    S. Karlin, Total Positivity, Stanford University Press (Stanford), 1968.Google Scholar
  16. [16]
    H.L. Montgomery and R.C. Vaughan, Mean values of character sums, Canadian J. Math. 31 (1979), 476–487.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    G. Pólya, Verschiedene Bemerkung zur Zahlentheorie, Jber. Deutsch. Math. Verein 28 (1919), 31–40.MATHGoogle Scholar
  18. [18]
    G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vols I and II, Springer-Verlag (Berlin ), 1964.MATHGoogle Scholar
  19. [19]
    D. Wolke, Eine Bemerkung über das Legendre-Symbol, Monat. Math. 77 (1973), 267 - 275.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Bikhäuser Boston 1990

Authors and Affiliations

  • R. C. Baker
    • 1
  • Hugh L. Montgomery
    • 2
  1. 1.Royal Holloway and Bedford New CollegeEgham, SurreyEngland, UK
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations