EVALUATION OF THE CONVOLUTION SUM Sigma(al+bm=n) sigma(l)sigma(m) FOR (a, b) = (1, 48), (3, 16), (1, 54), (2, 27)

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Alaca S., Kesicioglu Y.

FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI, vol.61, no.1, pp.27-45, 2019 (ESCI)

• Publication Type: Article / Article
• Volume: 61 Issue: 1
• Publication Date: 2019
• Doi Number: 10.7169/facm/1742
• Journal Name: FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI
• Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, MathSciNet, zbMATH
• Page Numbers: pp.27-45
• Keywords: convolution sums, sum of divisors function, Eisenstein series, Eisenstein forms, modular forms, cusp forms, Dedekind eta function, octonary quadratic forms, representations
• Recep Tayyip Erdoğan University Affiliated: Yes

Abstract

We determine the convolution sum Sigma(al+bm=n) sigma(l)sigma(m) for (a, b) = (1, 48), (3, 16), (1, 54), (2, 27) 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 k(x(1)(2) +x(1)x(2) + x(2)(2) + x(3)(2)+ x(3)x(4) + x(4)(2)) + l(x(5)(2) + x(5)x(6) + x(6)(2) + x(7)(2) + x(7)x(8) + x(8)(2)) for (k, l) = (1, 16), (1, 18), (2, 9). A modular form approach is used.