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The convolution sum \(\sum_{m<n/8} \sigma(m) \sigma(n-8m)\). (English) Zbl 1130.11006

Let \(\sigma(n)\) the divisor function \(\sum_{d\mid n} d\) and \(W_k(n):=\sum_{1\leq m<{n\over k}} \sigma(m)\sigma(n- km)\). The convolution sum \(W_k(n)\) is known for \(k= 1,2,3,4,9\) and all \(n\in\mathbb N\). The author evaluates in this paper the sum \(W_8(n)\). He applied his result to determine the number of representations of \(n\) by the quadratic form \(x^2_1+ x^2_2+ x^2_3+ x^2_4+ 2(x^2_5+ x^2_6+ x^2_7+ x^2_8)\).

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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