Evaluation of the convolution sums Sigma(l+27m=n) sigma(l)sigma(m) and Sigma(l+32m=n) sigma(l)sigma(m)

Alaca S., Kesicioglu Y.

INTERNATIONAL JOURNAL OF NUMBER THEORY, vol.12, no.1, pp.1-13, 2016 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 12 Issue: 1
  • Publication Date: 2016
  • Doi Number: 10.1142/s1793042116500019
  • Journal Name: INTERNATIONAL JOURNAL OF NUMBER THEORY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1-13
  • Keywords: Convolution sums, sum of divisors function, Eisenstein series, Eisenstein forms, modular forms, cusp forms, Dedekind eta function, octonary quadratic forms, representations, QUADRATIC-FORMS, REPRESENTATIONS, NUMBERS
  • Recep Tayyip Erdoğan University Affiliated: Yes

Abstract

We determine the convolution sums Sigma(l+27m= n) sigma(l)sigma(m) and Sigma(l+32m= n) sigma(l)sigma(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 x(1)(2) + x(1)x(2) + x(2)(2) + x(3)(3) + x(3)x(4) + x(4)(2) + 9(x(5)(2) + x(5)x(6) + x(6)(2) + x(7)(2) + x(7)x(8) + x(8)(2)) and x(1)(2) + x(2)(2) + x(3)(2) + x(4)(2) + 8(x(5)(2) + x(6)(2) + x(7)(2) + x(8)(2)). A modular form approach is used.