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Oscillations of Quadratic L-Functions

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Analytic Number Theory

Part of the book series: Progress in Mathematics ((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)}.

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References

  1. R.C. Baker and G. Harman, Unbalanced quadratic residues and non-residues, Proc. Camb. Philos. Soc. 98 (1975), 9–17.

    Article  MathSciNet  Google Scholar 

  2. P.T. Bateman, G.B. Purdy and S. Wagstaff, Some numerical results on Fekete polynomials, Math. Comp. 29 (1975), 7–23.

    Article  MathSciNet  MATH  Google Scholar 

  3. S.D. Chowla, Note on Dirichlet’s L-fucnctions, Acta Arith. 1 (1936), 113–114.

    MathSciNet  Google Scholar 

  4. K.L Chung, A course in probability theory, Harcourt, Brace and World (New York ), 1968.

    MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  6. H. Davenport, Multiplicative Number Theory, Second Edition, Springer-Verlag (New York), 1980, 178 pp.

    MATH  Google Scholar 

  7. P.D.T.A. Elliott, On the mean value of f(p), Proc. London Math. Soc. (3) 21 (1970), 28–96.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Heilbronn, On real characters, Acta Arith. 2 (1937), 212–213.

    MATH  Google Scholar 

  10. M. Jutila, On character sums and class numbers, J. Number Theory 5 (1973), 203–214.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Jutila, A density estimate for L-functions with a real character, Ann. Acad. Sci. Fenn. Ser. AI 508 (1972), 10 pp.

    MathSciNet  Google Scholar 

  12. M. Jutila, On the mean values of L-functions and short character sums with real characters, Acta Arith. 26 (1975), 405–410.

    MathSciNet  MATH  Google Scholar 

  13. M. Jutila, On the mean values of Dirichlet polynomials with real characters, Acta Arith. 27 (1975), 191–198.

    MathSciNet  MATH  Google Scholar 

  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. S. Karlin, Total Positivity, Stanford University Press (Stanford), 1968.

    Google Scholar 

  16. H.L. Montgomery and R.C. Vaughan, Mean values of character sums, Canadian J. Math. 31 (1979), 476–487.

    Article  MathSciNet  MATH  Google Scholar 

  17. G. Pólya, Verschiedene Bemerkung zur Zahlentheorie, Jber. Deutsch. Math. Verein 28 (1919), 31–40.

    MATH  Google Scholar 

  18. G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vols I and II, Springer-Verlag (Berlin ), 1964.

    MATH  Google Scholar 

  19. D. Wolke, Eine Bemerkung über das Legendre-Symbol, Monat. Math. 77 (1973), 267 - 275.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Royal Holloway and Bedford New College, Egham, Surrey, TW 20 OEX, England, UK

    R. C. Baker

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1003, USA

    Hugh L. Montgomery

Authors
  1. R. C. Baker
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  2. Hugh L. Montgomery
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 west Green Street, 61801, Urbana, Illinois, USA

    Bruce C. Berndt , Harold G. Diamond , Heini Halberstam  & Adolf Hildebrand , ,  & 

Additional information

Dedicated to Paul T. Bateman on his seventieth birthday

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© 1990 Bikhäuser Boston

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Baker, R.C., Montgomery, H.L. (1990). Oscillations of Quadratic L-Functions. In: Berndt, B.C., Diamond, H.G., Halberstam, H., Hildebrand, A. (eds) Analytic Number Theory. Progress in Mathematics, vol 85. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3464-7_3

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  • DOI: https://doi.org/10.1007/978-1-4612-3464-7_3

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-3481-0

  • Online ISBN: 978-1-4612-3464-7

  • eBook Packages: Springer Book Archive

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