References
L. V. Ahlfors,Complex Analysis, 2nd edition, McGraw-Hill, New York, 1966.
R. P. Boas,Entire Functions, Academic Press, New York, 1954.
E. Bombieri and D. A. Hejhal,Sur les zéros des fonctions zÊta d’Epstein, C.R. Acad. Sci. Paris304 (1987), 213–217.
N. G. DeBruijn,The roots of trigonometric integrals, Duke Math. J.17 (1950), 197–226.
M. Deuring,Zetafunktionen quadratischer Formen, J. Reine Angew. Math.172 (1935), 226–252.
M. Deuring,ImaginÄre quadratische Zahlkörper mit der Klassenzahl 1, Math. Z.37 (1933), 405–415.
A. Erdélyi et al.,Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953.
A. Erdélyi et al.,Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.
A. Gray and G. B. Mathews,A Treatise on Bessel Functions and Their Applications to Physics, 2nd edition, McMillan, London, 1922.
E. Hecke,Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1959.
D. A. Hejhal,Zeros of Epstein zeta functions and supercomputers, inProc. International Congress of Mathematicians, Berkeley, 1986, pp. 1362–1384.
D. A. Hejhal,The Selberg Trace Formula for PSL(2, ℝ), Vol. 1, Springer Lecture Notes548 (1976).
D. A. Hejhal,The Selberg Trace Formula for PSL(2, ℝ), Vol. 2, Springer Lecture Notes1001 (1983).
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).
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).
A. E. Ingham,The Distribution of Prime Numbers, Cambridge University Press, 1932.
J. Lagarias and A. Odlyzko,On computing Artin L-functions in the critical strip, Math. Comp.33 (1979), 1081–1095, especially §3.
E. Landau,Vorlesungen über Zahlentheorie, Vols. 1–3, S. Hirzel, Leipzig, 1927.
B. Ja. Levin,Distribution of Zeros of Entire Functions, Translations of Math. Monographs No. 5, Am. Math. Soc., 1964.
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).
C. Meyer,Die Berechnung der Klassenzahl abelscher Körper über quadratischen Zahlkörpern, Akademie-Verlag, Berlin, 1957.
R. Nevanlinna,Analytic Functions, Springer-Verlag, Berlin, 1970.
G. Pólya,Bemerkung über die Integraldarstellung der Riemannschen ξ-Funktion, Acta Math.48 (1926), 305–317.
G. Pólya,über trigonometrische Integrale mit nur reellen Nullstellen, J. Reine Angew. Math.158 (1927), 6–18.
G. Pólya,Collected Papers, Vol. 2, MIT Press, 1974.
G. Pólya and G. Szegö,Aufgaben und LehrsÄtze aus der Analysis, Vol. 1, Springer-Verlag, Berlin, 1925.
H. S. A. Potter and E. C. Titchmarsh,The zeros of Epstein’s zeta-functions, Proc. London Math. Soc.39 (1935), 372–384.
B. Riemann,Gesammelte Mathematische Werke, 2nd edition, B. G. Teubner, Leipzig, 1892.
C. L. Siegel,Advanced Analytic Number Theory, 2nd edition, Tata Inst. Fund. Research, Bombay, 1980.
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.
C. L. Siegel,Contributions to the theory of the Dirichlet L-series and the Epstein zeta functions, Ann. of Math.44 (1943), 143–172.
C. L. Siegel,Die Funktionalgleichungen einiger Dirichletscher Reihen, Math. Z.63 (1956), 363–373, especially p. 369(top) and eqs. (8)(18)(20)(26).
H. Stark,On the zeros of Epstein’s zeta functions, Mathematika14 (1967), 47–55.
H. Stark,Values of L-functions at s = 1,part I, Adv. in Math.7 (1971), 301–343.
E. C. Titchmarsh,The Theory of the Riemann Zeta-Function, Oxford University Press, 1951.
G. N. Watson,A Treatise on the Theory of Bessel Functions, 2nd edition, Cambridge University Press, 1944.
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Supported in part by NSF Grants DMS 86-07958, 89-10744.
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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
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DOI: https://doi.org/10.1007/BF02789198