Class number formulae of Dirichlet type

Authors:
Richard H. Hudson and Kenneth S. Williams

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

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Abstract | References | Similar Articles | Additional Information

Abstract: Applying a theorem of Johnson and Mitchell, some new class number formulae are derived.

- Bruce C. Berndt,
*Classical theorems on quadratic residues*, Enseign. Math. (2)**22**(1976), no. 3-4, 261–304. MR**441835**
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,". - 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," - 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 https://doi.org/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 https://doi.org/10.1007/BF02418573

*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.

*J. Reine Angew. Math.*, v. 127, 1904, pp. 1-19.

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Keywords:
Dirichlet type class number formulae,
class numbers of imaginary quadratic number fields

Article copyright:
© Copyright 1982
American Mathematical Society