In his monumental treatise Disquisitiones Arithmeticae, Gauß conjectured that the Class Number
of an Imaginary quadratic field with Discriminant tends to infinity with . A proof was finally given by Heilbronn (1934), and Siegel (1936) showed that for
any , there exists a constant such that
Goldfeld (1976) showed that if there exists a ``Weil curve'' whose associated Dirichlet L-Series has a zero of at least third order at , then for any , there exists an effectively computable
constant such that
See also Class Number, Gauss's Class Number Problem, Heegner Number
References
Arno, S.; Robinson, M. L.; and Wheeler, F. S. ``Imaginary Quadratic Fields with Small Odd Class Number.''
http://www.math.uiuc.edu/Algebraic-Number-Theory/0009/.
Böcherer, S. ``Das Gauß'sche Klassenzahlproblem.'' Mitt. Math. Ges. Hamburg 11, 565-589, 1988.
Gauss, C. F. Disquisitiones Arithmeticae. New Haven, CT: Yale University Press, 1966.
Goldfeld, D. M. ``The Class Number of Quadratic Fields and the Conjectures of Birch and Swinnerton-Dyer.''
Ann. Scuola Norm. Sup. Pisa 3, 623-663, 1976.
Gross, B. and Zaiger, D. ``Points de Heegner et derivées de fonctions .'' C. R. Acad. Sci. Paris 297, 85-87, 1983.
Heilbronn, H. ``On the Class Number in Imaginary Quadratic Fields.'' Quart. J. Math. Oxford Ser. 25, 150-160, 1934.
Oesterlé, J. ``Nombres de classes des corps quadratiques imaginaires.'' Astérique 121-122, 309-323, 1985.
Siegel, C. L. ``Über die Klassenzahl quadratischer Zahlkörper.'' Acta. Arith. 1, 83-86, 1936.
© 1996-9 Eric W. Weisstein