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Evaluating convolution sums of the divisor function by quasimodular forms. (English) Zbl 1142.11027
Let \(W_N(n)=\sum_{m<n/N}\sigma_1(m)\sigma_1(n-Nm)\), where \(\sigma_1\) is the usual sum of divisors function. Many authors including S. Ramanujan have evaluated certain values of \(W_N(n)\) and related convolution sums. The present author provides a systematic method based on quasimodular forms to evaluate these kind of sums. This extension of modular forms has been constructed by M. Kaneko and D. Zagier [“A generalized Jacobi theta function and quasimodular forms”, Prog. Math. 129, 165–172 (1995; Zbl 0892.11015)].

11A25 Arithmetic functions; related numbers; inversion formulas
11F11 Holomorphic modular forms of integral weight
11F20 Dedekind eta function, Dedekind sums
11F25 Hecke-Petersson operators, differential operators (one variable)
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