Some New Old-Fashioned Modular Identities

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.