## Abstract

This paper uses modular functions on the theta group to derive an exact formula for the sum 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.)

$$\sum\limits_{\left| j \right| \leqslant n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } {\sigma \left( {n - j^2 } \right)} $$

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