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, cilt.12, sa.1, ss.1-13, 2016 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 12 Sayı: 1
  • Basım Tarihi: 2016
  • Doi Numarası: 10.1142/s1793042116500019
  • Dergi Adı: INTERNATIONAL JOURNAL OF NUMBER THEORY
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.1-13
  • Anahtar Kelimeler: 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 Üniversitesi Adresli: Evet

Özet

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.