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Mathematics of Computation

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Class number formulae of Dirichlet type
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by Richard H. Hudson and Kenneth S. Williams PDF
Math. Comp. 39 (1982), 725-732 Request permission

Abstract:

Applying a theorem of Johnson and Mitchell, some new class number formulae are derived.
References
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  • James D. Currie & Kenneth S. Williams, "Class numbers and biquadratic reciprocity." (Submitted.) G. L. Dirichlet, "Recherches sur diverses applications de l’analyse infinitésimale à la théorie des nombres,". J. Reine Angew. Math., v. 19, 1839, pp. 324-369. G. L. Dirichlet, "Recherches sur diverses applications de l’analyse infinitésimale à la théorie des nombres, second partie,". J. Reine Angew. Math., v. 21, 1840, pp. 134-155. J. W. L. Glaisher, "On the expression for the number of classes of a negative determinant, and on the numbers of positives in the octants of P," Quart. J. Math., v. 34, 1903, pp. 178-204. H. Holden, "On various expressions for h, the number of properly primitive classes for a determinant $- p$, where p is a prime of the form $4n + 3$ (first paper)," Messenger Math., v. 35, 1905/1906, pp. 73-80. H. Holden, "On various expressions for h, the number of properly primitive classes for a determinant $- p$, where p is of the form $4n + 3$, and is a prime or the product of different primes (second paper)," Messenger Math., v. 35, 1905/1906, pp. 102-110. H. Holden, "On various expressions for h, the number of properly primitive classes for any negative determinant, not involving a square factor (third paper)," Messenger Math., v. 35, 1905/1906, pp. 110-117. H. Holden, "On various expressions for h, the number of properly primitive classes for a negative determinant (fourth paper)," Messenger Math., v. 36, 1906/1907, pp. 69-75. H. Holden, "On various expressions for h, the number of properly primitive classes for a determinant $- p$, where p is of the form $4n + 3$, and is a prime or the product of different primes (addition to the second paper)," Messenger Math., v. 36, 1906/1907, pp. 75-77. H. Holden, "On various expressions for h, the number of properly primitive classes for a negative determinant not containing a square factor (fifth paper)," Messenger Math., v. 36, 1906/1907, pp. 126-134. H. Holden, "On various expressions for h, the number of properly primitive classes for any negative determinant, not containing a square factor (sixth paper)," Messenger Math., v. 37, 1907/1908, pp. 13-16.
  • Wells Johnson and Kevin J. Mitchell, Symmetries for sums of the Legendre symbol, Pacific J. Math. 69 (1977), no. 1, 117–124. MR 434936
  • Louis C. Karpinski, "Über die Verteilung der quadratischen Reste," J. Reine Angew. Math., v. 127, 1904, pp. 1-19.
  • M. Lerch, Essais sur le calcul du nombre des classes de formes quadratiques binaires aux coefficients entiers, Acta Math. 29 (1905), no. 1, 333–424 (French). MR 1555020, DOI 10.1007/BF02403208
  • M. Lerch, Essais sur le calcul du nombre des classes de formes quadratiques binaires aux coefficients entiers, Acta Math. 30 (1906), no. 1, 203–293 (French). MR 1555029, DOI 10.1007/BF02418573
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Math. Comp. 39 (1982), 725-732
  • MSC: Primary 12A50; Secondary 12A25
  • DOI: https://doi.org/10.1090/S0025-5718-1982-0669664-9
  • MathSciNet review: 669664