We determine the convolution sums Σl+27m=n ω(l)ω(m) and Σl+32m=n ω(l)ω(m) for all positive integers n. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of n by the octonary quadratic forms x12 + x 1x2 + x22 + x 32 + x 3x4 + x42 + 9(x 52 + x 5x6 + x62 + x 72 + x 7x8 + x82) and x12 + x 22 + x 32 + x 42 + 8(x 52 + x 62 + x 72 + x 82). A modular form approach is used.

Additional Metadata
Keywords Convolution sums, cusp forms, Dedekind eta function, Eisenstein forms, Eisenstein series, modular forms, octonary quadratic forms, representations, sum of divisors function
Persistent URL dx.doi.org/10.1142/S1793042116500019
Journal International Journal of Number Theory
Citation
Alaca, S, & Kesicioʇlu, Y. (Yavuz). (2016). Evaluation of the convolution sums ∑l+27m=nσ(l)σ(m) and ∑l+32m=nσ(l)σ(m). International Journal of Number Theory, 12(1), 1–13. doi:10.1142/S1793042116500019