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
Article

Abstract

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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References

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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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