Some new old-fashioned modular identities. (English) Zbl 0909.11018

Bateman, Paul T.; Knopp, Marvin I.

Ramanujan J. 2, No. 1-2, 247-269 (1998).

The paper uses modular functions on the theta group to derive an exact formula for the sum \[ \sum_{| j|\leq\sqrt u}\sigma(u-j^2) \] in terms of the singular series for the number of representations of an integer as a sum of 5 squares (here \(\sigma(k)\) denotes the sum of the divisors of \(k\) if \(k\) is a positive integer and \(\sigma(0) =-{1\over 24})\). Several related identities are derived and discussed.

Reviewer: M.Peters (Münster)

MSC:

11F11	Holomorphic modular forms of integral weight
11E25	Sums of squares and representations by other particular quadratic forms
11A25	Arithmetic functions; related numbers; inversion formulas

Keywords:

modular identities; sums of squares; sum of divisors; modular functions; theta group