The convolution sum, (equation presented) where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms a(x1 2+x2 2+x3 2+x4 2)+b(x5 2+x6 2+x7 2+x8 2), where (a, b) = (1, 11), (1, 13).

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Keywords Convolution Sums, Cusp Forms, Dedekind eta function, Eisenstein Series, Modular Forms, Number of Representations, Octonary quadratic Forms, Sums of Divisors function
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Journal Open Mathematics
Ntienjem, E. (Ebénézer). (2017). Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52. Open Mathematics, 15(1), 446–458. doi:10.1515/math-2017-0041