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.

**[1]**Bruce C. Berndt,*Classical theorems on quadratic residues*, Enseignement Math. (2)**22**(1976), no. 3–4, 261–304. MR**0441835****[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. 324-369.**[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. 134-155.**[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. 178-204.**[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. 73-80.**[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. 102-110.**[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. 110-117.**[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. 69-75.**[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. 75-77.**[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. 126-134.**[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. 13-16.**[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****[14]**Louis C. Karpinski, "Über die Verteilung der quadratischen Reste,"*J. Reine Angew. Math.*, v. 127, 1904, pp. 1-19.**[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**, https://doi.org/10.1007/BF02403208**[16]**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**, https://doi.org/10.1007/BF02418573

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669664-9

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

Article copyright:
© Copyright 1982
American Mathematical Society