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Convolution sums involving the divisor function. (English) Zbl 1156.11301

Summary: The series \(L_{r,4}(q)=\sum_{n=0}^\infty\sigma(4n+r)q^{4n+r},\) \(M_{r,4}(q)=\sum_{n=0}^\infty\sigma_3(4n+r)q^{4n+r},\) \(N_{r,4}(q)=\sum_{n=0}^\infty\sigma_5(4n+r)q^{4n+r},\) \(r=0,1,2,3, \) are evaluated and used to prove convolution formulae such as \[ \sum_{m\leq n}\sigma(4m-3)\sigma(4n-(4m-3))=4\sigma_3(n)-4\sigma_3(\tfrac12n). \]

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11F11 Holomorphic modular forms of integral weight
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