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The convolution sum \(\sum_{m<n/9}\sigma(m)\sigma(n-9m)\). (English) Zbl 1082.11003

Let \(\sigma_j(n)= \sum_{d|n} d^j\) and \(W_k(n)= \sum_{0< m<{n\over k}}\sigma(m) \sigma(n- km)\). Liouville has proved \(W_1(n)= {5\over 12}\sigma_3(n)+ {1- 6n\over 12}\sigma_1(n)\) for all positive integers \(n\). Similar but more complicated formulae are known for \(k= 2,3,4,5\) and \(9\). The author proves a formula for \(W_9(n)\), which contains the known one. He uses Eisenstein-series, Ramanujan’s tau-function and their relations.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11E25 Sums of squares and representations by other particular quadratic forms
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