Journal d’Analyse Mathematique

, Volume 55, Issue 1, pp 59–95 | Cite as

On a result of G. Pólya concerning the Riemann ξ-function

  • Dennis A. Hejhal
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. V. Ahlfors,Complex Analysis, 2nd edition, McGraw-Hill, New York, 1966.MATHGoogle Scholar
  2. 2.
    R. P. Boas,Entire Functions, Academic Press, New York, 1954.MATHGoogle Scholar
  3. 3.
    E. Bombieri and D. A. Hejhal,Sur les zéros des fonctions zÊta d’Epstein, C.R. Acad. Sci. Paris304 (1987), 213–217.MATHMathSciNetGoogle Scholar
  4. 4.
    N. G. DeBruijn,The roots of trigonometric integrals, Duke Math. J.17 (1950), 197–226.CrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Deuring,Zetafunktionen quadratischer Formen, J. Reine Angew. Math.172 (1935), 226–252.MATHGoogle Scholar
  6. 6.
    M. Deuring,ImaginÄre quadratische Zahlkörper mit der Klassenzahl 1, Math. Z.37 (1933), 405–415.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. Erdélyi et al.,Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953.Google Scholar
  8. 8.
    A. Erdélyi et al.,Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.Google Scholar
  9. 9.
    A. Gray and G. B. Mathews,A Treatise on Bessel Functions and Their Applications to Physics, 2nd edition, McMillan, London, 1922.MATHGoogle Scholar
  10. 10.
    E. Hecke,Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1959.MATHGoogle Scholar
  11. 11.
    D. A. Hejhal,Zeros of Epstein zeta functions and supercomputers, inProc. International Congress of Mathematicians, Berkeley, 1986, pp. 1362–1384.Google Scholar
  12. 12.
    D. A. Hejhal,The Selberg Trace Formula for PSL(2, ℝ), Vol. 1, Springer Lecture Notes548 (1976).Google Scholar
  13. 13.
    D. A. Hejhal,The Selberg Trace Formula for PSL(2, ℝ), Vol. 2, Springer Lecture Notes1001 (1983).Google Scholar
  14. 14.
    D. A. Hejhal,Roots of quadratic congruences and eigenvalues of the non-Euclidean Laplacian, inThe Selberg Trace Formula and Related Topics, D. Hejhal, P. Sarnak and A. Terras (eds.), Contemporary Mathematics Vol. 53, Am. Math. Soc., 1986, pp. 277–339, especially (3.2)(7.7)(7.11).Google Scholar
  15. 15.
    D. A. Hejhal,Some Dirichlet series with coefficients related to periods of automorphic eigenforms, Proc. Japan Acad.58A (1982), 413–417 and59A (1983), 335–338, especially theorem 1 and equations (10.2)–(10.5).MathSciNetGoogle Scholar
  16. 16.
    A. E. Ingham,The Distribution of Prime Numbers, Cambridge University Press, 1932.Google Scholar
  17. 17.
    J. Lagarias and A. Odlyzko,On computing Artin L-functions in the critical strip, Math. Comp.33 (1979), 1081–1095, especially §3.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    E. Landau,Vorlesungen über Zahlentheorie, Vols. 1–3, S. Hirzel, Leipzig, 1927.MATHGoogle Scholar
  19. 19.
    B. Ja. Levin,Distribution of Zeros of Entire Functions, Translations of Math. Monographs No. 5, Am. Math. Soc., 1964.Google Scholar
  20. 20.
    H. Maass,Konstruktion ganzer Modulformen halbzahliger Dimension mit θ-Multiplikatoren in einer und zwei Variabeln, Abh. Math. Sem. Hamburg12 (1938), 133–162, especially (2)(8)(14).Google Scholar
  21. 21.
    C. Meyer,Die Berechnung der Klassenzahl abelscher Körper über quadratischen Zahlkörpern, Akademie-Verlag, Berlin, 1957.MATHGoogle Scholar
  22. 22.
    R. Nevanlinna,Analytic Functions, Springer-Verlag, Berlin, 1970.MATHGoogle Scholar
  23. 23.
    G. Pólya,Bemerkung über die Integraldarstellung der Riemannschen ξ-Funktion, Acta Math.48 (1926), 305–317.CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    G. Pólya,über trigonometrische Integrale mit nur reellen Nullstellen, J. Reine Angew. Math.158 (1927), 6–18.MATHGoogle Scholar
  25. 25.
    G. Pólya,Collected Papers, Vol. 2, MIT Press, 1974.Google Scholar
  26. 26.
    G. Pólya and G. Szegö,Aufgaben und LehrsÄtze aus der Analysis, Vol. 1, Springer-Verlag, Berlin, 1925.MATHGoogle Scholar
  27. 27.
    H. S. A. Potter and E. C. Titchmarsh,The zeros of Epstein’s zeta-functions, Proc. London Math. Soc.39 (1935), 372–384.MATHCrossRefGoogle Scholar
  28. 28.
    B. Riemann,Gesammelte Mathematische Werke, 2nd edition, B. G. Teubner, Leipzig, 1892.MATHGoogle Scholar
  29. 29.
    C. L. Siegel,Advanced Analytic Number Theory, 2nd edition, Tata Inst. Fund. Research, Bombay, 1980.MATHGoogle Scholar
  30. 30.
    C. L. Siegel,über Riemanns Nachlass zur analytischen Zahlentheorie, Quell, und Stud. zur Geschichte der Math. Astr. Phys.2 (1932), 45–80;Gesammelte Abhandlungen, Vol. 1, Springer-Verlag, Berlin, 1966, pp. 275–310.Google Scholar
  31. 31.
    C. L. Siegel,Contributions to the theory of the Dirichlet L-series and the Epstein zeta functions, Ann. of Math.44 (1943), 143–172.CrossRefMathSciNetGoogle Scholar
  32. 32.
    C. L. Siegel,Die Funktionalgleichungen einiger Dirichletscher Reihen, Math. Z.63 (1956), 363–373, especially p. 369(top) and eqs. (8)(18)(20)(26).MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    H. Stark,On the zeros of Epstein’s zeta functions, Mathematika14 (1967), 47–55.MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    H. Stark,Values of L-functions at s = 1,part I, Adv. in Math.7 (1971), 301–343.MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    E. C. Titchmarsh,The Theory of the Riemann Zeta-Function, Oxford University Press, 1951.Google Scholar
  36. 36.
    G. N. Watson,A Treatise on the Theory of Bessel Functions, 2nd edition, Cambridge University Press, 1944.Google Scholar

Copyright information

© Hebrew University of Jerusalem 1990

Authors and Affiliations

  • Dennis A. Hejhal
    • 1
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

Personalised recommendations