Celebratio Mathematica — Bateman

Sums of squares and modular number theory

Works connected to Paul Erdős

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P. T. Bateman and M. I. Knopp: “Some new old-fashioned modular identities,” Ramanujan J. 2 : 1–2 (1998), pp. 247–269. Dedicated to the memory of Paul Erdős. MR 1642881 Zbl 0909.11018 article

This paper uses modular functions on the theta group to derive an exact formula for the sum \[ \sum_{|j|\leq n^{1/2}}\sigma(n - j^2) \] in terms of the singular series for the number of representations of an integer as a sum of five squares. (Here \( \sigma(k) \) denotes the sum of the divisors of \( k \) if \( k \) is a positive integer and \( \sigma(0) = -1/24 \).)

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.

Marvin Isadore Knopp	Related
Paul Erdős	Related