The Ramanujan Journal

, Volume 2, Issue 1–2, pp 247–269 | Cite as

Some New Old-Fashioned Modular Identities

  • Paul T. Bateman
  • Marvin I. Knopp


This paper uses modular functions on the theta group to derive an exact formula for the sum
$$\sum\limits_{\left| j \right| \leqslant n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } {\sigma \left( {n - j^2 } \right)} $$
in terms of the singular series for the number of representations of an integer as a sum of five squares. (Here σ(k) denotes the sum of the divisors of k if k is a positive integer and σ(0) =-1/24.)

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.

modular forms sum-of-divisors function theta group sums of squares singular series 


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  1. 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. 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. 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. 4.
    Leonard E. Dickson, Studies in the Theory of Numbers, University of Chicago Press, 1930.Google Scholar
  5. 5.
    T. Estermann, “On the representations of a number as a sum of squares,” Acta Arith. 2 (1936), 47–79.Google Scholar
  6. 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. 7.
    Marvin I. Knopp, Modular Functions in Analytic Number Theory, Markham, 1970 or Chelsea, 1993.Google Scholar
  8. 8.
    Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, 1993.Google Scholar
  9. 9.
    Hans Rademacher, Topics in Analytic Number Theory, Springer-Verlag, 1973.Google Scholar
  10. 10.
    Robert A. Rankin, Modular Forms and Functions, Cambridge University Press, 1977.Google Scholar
  11. 11.
    Arnold Walfisz, Gitterpunkte in mehrdimensionalen Kugeln, Monografie Matematyczne, Tom 33, 1957.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Paul T. Bateman
    • 1
  • Marvin I. Knopp
    • 2
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbana
  2. 2.Department of MathematicsTemple UniversityPhiladelphia

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