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Some New Old-Fashioned Modular Identities

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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.

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References

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois, 61801-2975. E-mail

    Paul T. Bateman

  2. Department of Mathematics, Temple University, Philadelphia, Pennsylvania, 19122-2585

    Marvin I. Knopp

Authors
  1. Paul T. Bateman
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  2. Marvin I. Knopp
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Bateman, P.T., Knopp, M.I. Some New Old-Fashioned Modular Identities. The Ramanujan Journal 2, 247–269 (1998). https://doi.org/10.1023/A:1009782529605

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