On the -rank of quadratic fields and the Euler product

Authors:
Carol Neild and Daniel Shanks

Journal:
Math. Comp. **28** (1974), 279-291

MSC:
Primary 12A25; Secondary 12A65

DOI:
https://doi.org/10.1090/S0025-5718-1974-0352042-5

MathSciNet review:
0352042

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Abstract: This paper covers many (closely related) topics: the distribution of the 3-Sylow subgroups of imaginary quadratic fields; the possibility of finding 3-ranks greater than 4; some questions concerning ; and the convergence of Euler products and its relation to the extended Riemann hypothesis. Two programs that were used in this investigation are described.

**[1]**Daniel Shanks & Peter Weinberger, "A quadratic field of prime discriminant requiring three generators for its class group, and related theory,"*Sierpiński Memorial Volume, Acta Arith.*, v. 21, 1972, pp. 71-87. MR**0309899 (46:9003)****[2]**Daniel Shanks, "New types of quadratic fields having three invariants divisible by 3,"*J. Number Theory*, v. 4, 1972, pp. 537-556. MR**0313220 (47:1775)****[3]**Daniel Shanks & Richard Serafin, "Quadratic fields with four invariants divisible by 3,"*Math. Comp.*, v. 27, 1973, pp. 183-187. Corrigendum,*ibid.*, p. 1012. MR**0330097 (48:8436a)****[4]**Peter Roquette, "On class field towers,"*Algebraic Number Theory*, (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 231-249. MR**36**#1418. MR**0218331 (36:1418)****[5]**H. Davenport & H. Heilbronn, "On the density of discriminants of cubic fields,"*Bull. London Math. Soc.*, v. 1, 1969, pp. 345-348. MR**40**#7223. MR**0254010 (40:7223)****[6]**H. Davenport & H. Heilbronn, "On the density of discriminants of cubic fields. II,"*Proc. Roy. Soc. London Ser. A*, v. 322, 1971, pp. 405-420. MR**0491593 (58:10816)****[7]**Daniel Shanks, "Class number, a theory of factorization, and genera,"*Proc. Sympos. Pure Math.*, vol. 20, Amer. Math. Soc., Providence, R.I., 1971, pp. 415-440. MR**0316385 (47:4932)****[8]**A. Scholz, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander,"*Crelle's J.*, v. 166, 1932, pp. 201-203.**[9]**L. J. Mordell,*Diophantine Equations*, Pure and Appl. Math., vol. 30, Academic Press, New York and London, 1969, Chapter 26. MR**40**#2600. MR**0249355 (40:2600)****[10]**G. Gras, "Extensions abéliennes non ramifiées de degré premier d'un corps quadratique,"*Bull. Soc. Math. France*, v. 100, 1972, pp. 177-193. MR**0302604 (46:1748)****[11]**Carol C. Neild,*SPEEDY, A Code for Estimating the Euler Product of a Dirichlet L Function*, CMD-8-73, 1973, Naval Ship R&D Center, Bethesda, Maryland.**[12]**Daniel Shanks, "Five number-theoretic algorithms,"*Proceedings of the Manitoba Conference on Numerical Mathematics, 1972*, University of Manitoba, Winnipeg, Canada, 1973. MR**0371855 (51:8072)****[13]**Richard H. Serafin,*Two Subroutines for the Solution of**(modulo N) and*(*modulo P*)*and their Applications*, CMD-7-73, 1973, Naval Ship R&D Center, Bethesda, Maryland.**[14]**Carol C. Neild,*CUROID, A Code for Computing the Cube Roots of the Identity of the Class Group of an Imaginary Quadratic Field*, CMD-9-73, 1973, Naval Ship R&D Center, Bethesda, Maryland.**[15]**Daniel Shanks, "The infrastructure of a real quadratic field and its applications,"*Proceedings of the 1972 Number Theory Conference*, University of Colorado, Boulder, Colorado, 1973, pp. 217-224. MR**0389842 (52:10672)**

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DOI:
https://doi.org/10.1090/S0025-5718-1974-0352042-5

Article copyright:
© Copyright 1974
American Mathematical Society