Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Small class numbers and extreme values of $ L$-functions of quadratic fields


Author: Duncan A. Buell
Journal: Math. Comp. 31 (1977), 786-796
MSC: Primary 12A25; Secondary 12A30, 12A70, 12-04
DOI: https://doi.org/10.1090/S0025-5718-1977-0439802-X
MathSciNet review: 0439802
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The table of class numbers h of imaginary quadratic fields described in [1] was placed on magnetic tape. This tape was then processed to find the occurrences of $ h \leqslant 125$ and to find the successive extreme values of the Dirichlet L-functions $ L(1,{\chi _{ - D}})$, $ {\chi _{ - D}}$ the Kronecker symbol of the field $ Q(\sqrt { - D} )$ of discriminant - D. A comparison was made between the observed extrema and the bounds obtained for the L-functions by Littlewood [5] assuming Riemann hypotheses.


References [Enhancements On Off] (What's this?)

  • [1] DUNCAN A. BUELL, "Class groups of quadratic fields," Math. Comp., v. 30, 1976, pp. 610-623. MR 0404205 (53:8008)
  • [2] D. SHANKS, Review of Richard B. Lakein and Sigekatu Kuroda, "Tables of class numbers $ h( - p)$ for fields $ Q(\surd - p)$, $ p \leqslant 465071$," UMT 39, Math. Comp., v. 24, 1970, pp. 491-493.
  • [3] DANIEL SHANKS, Systematic Examination of Littlewood's Bounds on $ L(1,\chi )$, Proc. Sympos. Pure Math., vol. 24, Amer. Math. Soc., Providence, R.I., 1973, pp. 267-283. MR 49 #2596. MR 0337827 (49:2596)
  • [4] D. H. LEHMER, EMMA LEHMER & DANIEL SHANKS, "Integer sequences having prescribed quadratic character," Math. Comp., v. 24, 1970, pp. 433-451. MR 42 #5889. MR 0271006 (42:5889)
  • [5] J. E. LITTLEWOOD, "On the class-number of the corpus $ P(\sqrt { - k} )$," Proc. London Math. Soc., v. 28, 1928, pp. 358-372.
  • [6] G. B. MATHEWS, Theory of Numbers, 2nd ed., Chelsea, New York, 1961. MR 23 #A3698. MR 0126402 (23:A3698)
  • [7] DANIEL SHANKS, "Calculation and applications of Epstein zeta functions," Math. Comp., v. 29, 1975, pp. 271-287. MR 0409357 (53:13114a)
  • [8] DANIEL SHANKS, "On Gauss's cldass number problems," Math. Comp., v. 23, 1969, pp. 151-163. MR 41 #6814. MR 0262204 (41:6814)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 12A25, 12A30, 12A70, 12-04

Retrieve articles in all journals with MSC: 12A25, 12A30, 12A70, 12-04


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1977-0439802-X
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society