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 and Peter Weinberger,*A quadratic field of prime discriminant requiring three generators for its class group, and related theory*, Acta Arith.**21**(1972), 71–87. MR**309899**, https://doi.org/10.4064/aa-21-1-71-87**[2]**Daniel Shanks,*New types of quadratic fields having three invariants divisible by 3*, J. Number Theory**4**(1972), 537–556. MR**313220**, https://doi.org/10.1016/0022-314X(72)90027-3**[3]**Daniel Shanks and Richard Serafin,*Quadratic fields with four invariants divisible by 3*, Math. Comp.**27**(1973), 183–187. MR**330097**, https://doi.org/10.1090/S0025-5718-1973-0330097-0**[4]**Peter Roquette,*On class field towers*, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 231–249. MR**0218331****[5]**H. Davenport and H. Heilbronn,*On the density of discriminants of cubic fields*, Bull. London Math. Soc.**1**(1969), 345–348. MR**254010**, https://doi.org/10.1112/blms/1.3.345**[6]**H. Davenport and H. Heilbronn,*On the density of discriminants of cubic fields. II*, Proc. Roy. Soc. London Ser. A**322**(1971), no. 1551, 405–420. MR**491593**, https://doi.org/10.1098/rspa.1971.0075**[7]**Daniel Shanks,*Class number, a theory of factorization, and genera*, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 415–440. MR**0316385****[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 Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR**0249355****[10]**Georges Gras,*Extensions abéliennes non ramifiées de degré premier d’un corps quadratique*, Bull. Soc. Math. France**100**(1972), 177–193 (French). MR**302604****[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 Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972) Utilitas Math., Winnipeg, Man., 1973, pp. 51–70. Congressus Numerantium, No. VII. MR**0371855****[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 Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972) Univ. Colorado, Boulder, Colo., 1972, pp. 217–224. MR**0389842**

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

Article copyright:
© Copyright 1974
American Mathematical Society