The Ramanujan Journal

, Volume 2, Issue 1–2, pp 247–269

# 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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