Several related identities are derived and discussed.
Two devices are used in the proofs. The first device establishes the equality of two expressions, neither of which is a modular form, by showing that the square of their difference is a modular form. The second device shows that a certain modular function is identically zero by noting that it has more zeros than poles in a fundamental region.
Unable to display preview. Download preview PDF.
- 1.P.T. Bateman, “On the representations of a number as the sum of three squares,” Trans. Amer. Math. Soc. 71 (1951), 70–101.Google Scholar
- 2.P.T. Bateman, “The asymptotic formula for the number of representations of an integer as a sum of five squares,” Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam, Birkhaüser, vol. 1, pp. 129–139, 1996.Google Scholar
- 3.Henri Cohen, “Sums involving the values at negative integers of L-functions of quadratic characters,” Math. Ann. 217 (1975), 271–285.Google Scholar
- 4.Leonard E. Dickson, Studies in the Theory of Numbers, University of Chicago Press, 1930.Google Scholar
- 5.T. Estermann, “On the representations of a number as a sum of squares,” Acta Arith. 2 (1936), 47–79.Google Scholar
- 6.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), 255–284; 29 (1927), 845–847.Google Scholar
- 7.Marvin I. Knopp, Modular Functions in Analytic Number Theory, Markham, 1970 or Chelsea, 1993.Google Scholar
- 8.Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, 1993.Google Scholar
- 9.Hans Rademacher, Topics in Analytic Number Theory, Springer-Verlag, 1973.Google Scholar
- 10.Robert A. Rankin, Modular Forms and Functions, Cambridge University Press, 1977.Google Scholar
- 11.Arnold Walfisz, Gitterpunkte in mehrdimensionalen Kugeln, Monografie Matematyczne, Tom 33, 1957.Google Scholar