Class number formulae of Dirichlet type
Authors:
Richard H. Hudson and Kenneth S. Williams
Journal:
Math. Comp. 39 (1982), 725732
MSC:
Primary 12A50; Secondary 12A25
MathSciNet review:
669664
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Applying a theorem of Johnson and Mitchell, some new class number formulae are derived.
 [1]
Bruce
C. Berndt, Classical theorems on quadratic residues,
Enseignement Math. (2) 22 (1976), no. 3–4,
261–304. MR 0441835
(56 #229)
 [2]
James D. Currie & Kenneth S. Williams, "Class numbers and biquadratic reciprocity." (Submitted.)
 [3]
G. L. Dirichlet, "Recherches sur diverses applications de l'analyse infinitésimale à la théorie des nombres,". J. Reine Angew. Math., v. 19, 1839, pp. 324369.
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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. 134155.
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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. 178204.
 [6]
H. Holden, "On various expressions for h, the number of properly primitive classes for a determinant , where p is a prime of the form (first paper)," Messenger Math., v. 35, 1905/1906, pp. 7380.
 [7]
H. Holden, "On various expressions for h, the number of properly primitive classes for a determinant , where p is of the form , and is a prime or the product of different primes (second paper)," Messenger Math., v. 35, 1905/1906, pp. 102110.
 [8]
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. 110117.
 [9]
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. 6975.
 [10]
H. Holden, "On various expressions for h, the number of properly primitive classes for a determinant , where p is of the form , and is a prime or the product of different primes (addition to the second paper)," Messenger Math., v. 36, 1906/1907, pp. 7577.
 [11]
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. 126134.
 [12]
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. 1316.
 [13]
Wells
Johnson and Kevin
J. Mitchell, Symmetries for sums of the Legendre symbol,
Pacific J. Math. 69 (1977), no. 1, 117–124. MR 0434936
(55 #7899)
 [14]
Louis C. Karpinski, "Über die Verteilung der quadratischen Reste," J. Reine Angew. Math., v. 127, 1904, pp. 119.
 [15]
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, http://dx.doi.org/10.1007/BF02403208
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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, http://dx.doi.org/10.1007/BF02418573
 [1]
 Bruce C. Berndt, "Classical theorems on quadratic residues," Enseign. Math., v. 22, 1976, pp. 261304. MR 0441835 (56:229)
 [2]
 James D. Currie & Kenneth S. Williams, "Class numbers and biquadratic reciprocity." (Submitted.)
 [3]
 G. L. Dirichlet, "Recherches sur diverses applications de l'analyse infinitésimale à la théorie des nombres,". J. Reine Angew. Math., v. 19, 1839, pp. 324369.
 [4]
 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. 134155.
 [5]
 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. 178204.
 [6]
 H. Holden, "On various expressions for h, the number of properly primitive classes for a determinant , where p is a prime of the form (first paper)," Messenger Math., v. 35, 1905/1906, pp. 7380.
 [7]
 H. Holden, "On various expressions for h, the number of properly primitive classes for a determinant , where p is of the form , and is a prime or the product of different primes (second paper)," Messenger Math., v. 35, 1905/1906, pp. 102110.
 [8]
 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. 110117.
 [9]
 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. 6975.
 [10]
 H. Holden, "On various expressions for h, the number of properly primitive classes for a determinant , where p is of the form , and is a prime or the product of different primes (addition to the second paper)," Messenger Math., v. 36, 1906/1907, pp. 7577.
 [11]
 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. 126134.
 [12]
 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. 1316.
 [13]
 Wells Johnson & Kevin J. Mitchell, "Symmetries for sums of the Legendre symbol," Pacific. J. Math.,v. 69, 1977, pp. 117124. MR 0434936 (55:7899)
 [14]
 Louis C. Karpinski, "Über die Verteilung der quadratischen Reste," J. Reine Angew. Math., v. 127, 1904, pp. 119.
 [15]
 M. Lerch, "Essais sur le calcul du nombre des classes de formes quadratiques binaires aux coefficients entiers," Acta Math., v. 29, 1905, pp. 333424. MR 1555020
 [16]
 M. Lerch, "Essais sur le calcul du nombre des classes de formes quadratiques binaires aux coefficients entiers," Acta Math., v. 30, 1906, pp. 203293. MR 1555029
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206696649
PII:
S 00255718(1982)06696649
Keywords:
Dirichlet type class number formulae,
class numbers of imaginary quadratic number fields
Article copyright:
© Copyright 1982
American Mathematical Society
