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XVI.—The Lattice Points of a Circle

Published online by Cambridge University Press:  15 September 2014

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Extract

1. A great deal of interest has of late years centred round the arithmetical function r(n), defined as the number of integer solutions of the equation

p2 + q2 = n,

so that, for example, r(1) = 4, the four solutions being (0, ±1) and (±1, 0).

Type
Proceedings
Copyright
Copyright © Royal Society of Edinburgh 1929

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References

Books and Papers Referred to

(1)I'a Bromwich, T. J., Theory of Infinite Series, London (2nd ed., 1926).Google Scholar
(2)van der Corput, J. G., “Zum Teilerproblem,” Math. Annalen, 98 (1928), pp. 697716.CrossRefGoogle Scholar
(3)van der Corput, J. G., “Zahlentheoretische Abschätzungen mit Anwendungen auf Gitterpunktprobleme (zweite Mitteilung),“ Math. Zeitschrift, 28 (1928), pp. 301310.CrossRefGoogle Scholar
(4)Hardy, G. H., “ Notes on Some Points in the Integral Calculus,” xxvii, Messenger of Math., 40 (1911), pp. 4451.Google Scholar
(5)Hardy, G. H., “ On Dirichlet's Divisor Problem,” Proc. London Math. Soc., (2), 15 (1916), pp. 125.Google Scholar
(6)Hardy, G. H., “ On the Expression of a Number as the Sum of Two Squares,” Quart. Journ. of Math., 46 (1915), pp. 263283.Google Scholar
(7)Landau, E., “ Computo asintotico dei nodi di un reticolato entro un cerchio,” Seminario matematico della facolta di science della R. Università di Roma, Rend, delle sedute dell' anno accademico, 19241925, (2), 3 (1926), pp. 3561.Google Scholar
(8)Landau, E., Vorlesungen über Zahlentheorie, Band 2, Teil viii, Leipzig (1927).Google Scholar
(9)Nieland, L. W., “Zum Kreisproblem,” Math. Annalen, 98 (1928). pp. 717736.CrossRefGoogle Scholar
(10)Voronoï, G., “ Sur le développement, a l'aide des fonctions cylindriques, des sommes doubles Σf(pm 2 + 2qmn + rn 2), oú pm 2 + 2qmn + rn 2 est une forme positive á coefficients entiers,” Verhandlungen des dritten int. Math.-Kongresses in Heidelberg, 1904 (1905), pp. 241245.Google Scholar
(11)Watson, G. N., Theory of Bessel Functions, Cambridge (1922).Google Scholar
(12)Weyl, H., “Über ein Problem aus dem Gebiet der Diophantischem Approximationen,” Göttinger Nachrichten (1914), pp. 234244.Google Scholar
(13)Weyl, H., “Über die Gleichverteilung von Zahlen mod Eins,” Math. Annalen, 77 (1916), pp. 313352.CrossRefGoogle Scholar
(14)Wilton, J. R., “Some Applications of a Transformation of Series,” Proc. London Math. Soc., (2), 27 (1927), pp. 81104.Google Scholar
(15)Wilton, J. R., “The Lattice Points of an n-dimensional Ellipsoid,” Journ. London Math. Soc., 2 (1927), pp. 227233.Google Scholar