On the representations of a number as the sum of three squares
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- by Paul T. Bateman
- Trans. Amer. Math. Soc. 71 (1951), 70-101
- DOI: https://doi.org/10.1090/S0002-9947-1951-0042438-4
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References
- Paul Bachmann Die analytische Zahlentheorie, Leipzig, 1894.
—Die Arithmetik von quadratischen Formen, part I, Leipzig, 1898.
P. T. Bateman On the representations of a number as the sum of three squares, Bull. Amer. Math. Soc. Abstract 52-9-262.
- S. Chowla, On the $k$-analogue of a result in the theory of the Riemann zeta function, Math. Z. 38 (1934), no. 1, 483–487. MR 1545462, DOI 10.1007/BF01170649 L. E. Dickson History of the theory of numbers, vol. 2, Washington, 1920. —Studies in the theory of numbers, Chicago, 1930. T. Estermann On the representations of a number as the sum of three or more products, Proc. London Math. Soc. (2) vol. 34 (1932) pp. 190-195. —On the representations of a number as a sum of squares, Acta Arithmetica vol. 2 (1936) pp. 47-79. L. R. Ford Automorphic functions, New York, 1929. G. H. Hardy On the representation of a number as the sum of any number of squares, and in particular of five or seven, Proc. London Math. Soc. (2) vol. 17 (1918) pp. xxii-xxiv (Records of proceedings at the meeting of March 14, 1918). —On the representation of a number as the sum of any number of squares, and in particular of five or seven, Proc. Nat. Acad. Sci. U.S.A. vol. 4 (1918) pp. 189-193.
- G. H. Hardy, On the representation of a number as the sum of any number of squares, and in particular of five, Trans. Amer. Math. Soc. 21 (1920), no. 3, 255–284. MR 1501144, DOI 10.1090/S0002-9947-1920-1501144-7 —Note on Ramanujan’s trigonometrical function ${C_q}(n)$, and certain series of arithmetical functions, Proc. Cambridge Philos. Soc. vol. 20 (1921) pp. 263-271. —Ramanujan, twelve lectures on subjects suggested by his life and work, Cambridge, 1940.
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125 Erich Hecke Darstellung von Klassenzahlen als Perioden von Integralen 3. Gattung aus dem Gebiet der elliptischen Modulfunktionen, Abh. Math. Sem. Hamburgischen Univ. vol. 4 (1925) pp. 211-223. —Theorie der Eisensteinschen Reihen höherer Stufe und ihre Anwendung auf Funktionen-theorie und Arithmetik, Abh. Math. Sem. Hamburgischen Univ. vol. 5 (1927) pp. 199-224.
- M. Kac, Almost periodicity and the representation of integers as sums of squares, Amer. J. Math. 62 (1940), 122–126. MR 841, DOI 10.2307/2371442 H. D. Kloostermann On the representations of numbers in the form $a{x^2} + b{y^2} + c{z^2} + d{t^2}$, Proc. London Math. Soc. (2) vol. 25 (1926) pp. 143-173.
- H. D. Kloosterman, On the representation of numbers in the form $ax^2+by^2+cz^2+dt^2$, Acta Math. 49 (1927), no. 3-4, 407–464. MR 1555249, DOI 10.1007/BF02564120 Edmund Landau Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Archiv der Mathematik und Physik (3) vol. 13 (1908) pp. 305-312. —Handbuch der Lehre von der Verteilung der Primzahlen, Leipzig, 1909. —Abschätzungen von Charaktersummen, Einheiten und Klassenzahlen, Nachr. Ges. Wiss. Göttingen (1918) pp. 79-97. —Vorlesungen über Zahlentheorie, vol. 1, Leipzig, 1927. Georg Landsberg Zur Theorie der Gausschen Summen und des linearen Transformationen der Thetafunktionen, J. Reine Angew. Math. vol. 111 (1893) pp. 234-253. R. Lipschitz Untersuchung einer aus vier Elementen gebildeten Reihe, J. Reine Angew. Math. vol. 54 (1857) pp. 313-328. L. J. Mordell On the representation of numbers as a sum of $2r$ squares, Quarterly Journal of Pure and Applied Mathematics vol. 48 (1917) pp. 93-104. —On the representations of a number as a sum of an odd number of squares, Transactions of the Cambridge Philosophical Society vol. 22 (1919) pp. 361-372. —Poisson’s summation formula and the Riemann zeta function, J. London Math. Soc. vol. 4 (1929) pp. 285-291. G. Pólya Über die Verteilung der quadratischen Reste und Nichtreste, Nachr. Ges. Wiss. Göttingen (1918) pp. 21-29. S. Ramanujan On certain trigonometrical sums and their applications in the theory of numbers, Transactions of the Cambridge Philosophical Society vol. 22 (1918) pp. 259-276. J. Schur Einige Bemerkungen zu der vorstehenden Arbeit des Herrn G. Pólya, Nachr. Ges. Wiss. Göttingen (1918) pp. 30-36.
- Carl Ludwig Siegel, Über die analytische Theorie der quadratischen Formen, Ann. of Math. (2) 36 (1935), no. 3, 527–606 (German). MR 1503238, DOI 10.2307/1968644 G. K. Stanley On the representations of a number as the sum of seven squares, J. London Math. Soc. vol. 2 (1927) pp. 91-96. E. T. Whittaker and E. M. Watson A course of modern analysis, Cambridge, 1927.
Bibliographic Information
- © Copyright 1951 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 71 (1951), 70-101
- MSC: Primary 10.0X
- DOI: https://doi.org/10.1090/S0002-9947-1951-0042438-4
- MathSciNet review: 0042438