Skip to main content
Log in

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

  • Published:
Journal d’Analyse Mathematique Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. L. V. Ahlfors,Complex Analysis, 2nd edition, McGraw-Hill, New York, 1966.

    MATH  Google Scholar 

  2. R. P. Boas,Entire Functions, Academic Press, New York, 1954.

    MATH  Google Scholar 

  3. E. Bombieri and D. A. Hejhal,Sur les zéros des fonctions zÊta d’Epstein, C.R. Acad. Sci. Paris304 (1987), 213–217.

    MATH  MathSciNet  Google Scholar 

  4. N. G. DeBruijn,The roots of trigonometric integrals, Duke Math. J.17 (1950), 197–226.

    Article  MathSciNet  Google Scholar 

  5. M. Deuring,Zetafunktionen quadratischer Formen, J. Reine Angew. Math.172 (1935), 226–252.

    MATH  Google Scholar 

  6. M. Deuring,ImaginÄre quadratische Zahlkörper mit der Klassenzahl 1, Math. Z.37 (1933), 405–415.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Erdélyi et al.,Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953.

    Google Scholar 

  8. A. Erdélyi et al.,Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.

    Google Scholar 

  9. A. Gray and G. B. Mathews,A Treatise on Bessel Functions and Their Applications to Physics, 2nd edition, McMillan, London, 1922.

    MATH  Google Scholar 

  10. E. Hecke,Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1959.

    MATH  Google Scholar 

  11. D. A. Hejhal,Zeros of Epstein zeta functions and supercomputers, inProc. International Congress of Mathematicians, Berkeley, 1986, pp. 1362–1384.

  12. D. A. Hejhal,The Selberg Trace Formula for PSL(2, ℝ), Vol. 1, Springer Lecture Notes548 (1976).

  13. D. A. Hejhal,The Selberg Trace Formula for PSL(2, ℝ), Vol. 2, Springer Lecture Notes1001 (1983).

  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).

  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).

    MathSciNet  Google Scholar 

  16. A. E. Ingham,The Distribution of Prime Numbers, Cambridge University Press, 1932.

  17. J. Lagarias and A. Odlyzko,On computing Artin L-functions in the critical strip, Math. Comp.33 (1979), 1081–1095, especially §3.

    Article  MATH  MathSciNet  Google Scholar 

  18. E. Landau,Vorlesungen über Zahlentheorie, Vols. 1–3, S. Hirzel, Leipzig, 1927.

    MATH  Google Scholar 

  19. B. Ja. Levin,Distribution of Zeros of Entire Functions, Translations of Math. Monographs No. 5, Am. Math. Soc., 1964.

  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. C. Meyer,Die Berechnung der Klassenzahl abelscher Körper über quadratischen Zahlkörpern, Akademie-Verlag, Berlin, 1957.

    MATH  Google Scholar 

  22. R. Nevanlinna,Analytic Functions, Springer-Verlag, Berlin, 1970.

    MATH  Google Scholar 

  23. G. Pólya,Bemerkung über die Integraldarstellung der Riemannschen ξ-Funktion, Acta Math.48 (1926), 305–317.

    Article  MathSciNet  MATH  Google Scholar 

  24. G. Pólya,über trigonometrische Integrale mit nur reellen Nullstellen, J. Reine Angew. Math.158 (1927), 6–18.

    MATH  Google Scholar 

  25. G. Pólya,Collected Papers, Vol. 2, MIT Press, 1974.

  26. G. Pólya and G. Szegö,Aufgaben und LehrsÄtze aus der Analysis, Vol. 1, Springer-Verlag, Berlin, 1925.

    MATH  Google Scholar 

  27. H. S. A. Potter and E. C. Titchmarsh,The zeros of Epstein’s zeta-functions, Proc. London Math. Soc.39 (1935), 372–384.

    Article  MATH  Google Scholar 

  28. B. Riemann,Gesammelte Mathematische Werke, 2nd edition, B. G. Teubner, Leipzig, 1892.

    MATH  Google Scholar 

  29. C. L. Siegel,Advanced Analytic Number Theory, 2nd edition, Tata Inst. Fund. Research, Bombay, 1980.

    MATH  Google Scholar 

  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. C. L. Siegel,Contributions to the theory of the Dirichlet L-series and the Epstein zeta functions, Ann. of Math.44 (1943), 143–172.

    Article  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  33. H. Stark,On the zeros of Epstein’s zeta functions, Mathematika14 (1967), 47–55.

    Article  MathSciNet  MATH  Google Scholar 

  34. H. Stark,Values of L-functions at s = 1,part I, Adv. in Math.7 (1971), 301–343.

    Article  MATH  MathSciNet  Google Scholar 

  35. E. C. Titchmarsh,The Theory of the Riemann Zeta-Function, Oxford University Press, 1951.

  36. G. N. Watson,A Treatise on the Theory of Bessel Functions, 2nd edition, Cambridge University Press, 1944.

Download references

Author information

Authors and Affiliations

Authors

Additional information

Supported in part by NSF Grants DMS 86-07958, 89-10744.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hejhal, D.A. On a result of G. Pólya concerning the Riemann ξ-function. J. Anal. Math. 55, 59–95 (1990). https://doi.org/10.1007/BF02789198

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02789198

Keywords

Navigation