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

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.

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
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