Evaluation of the convolution sum ∑al+bm=nσ(l)σ3(m) for (a,b)=(1,7),(7,1),(1,8),(8,1),(1,9),(9,1) and representations by certain quadratic forms in twelve variables

Alaca Ş., KESİCİOĞLU Y.

Indian Journal of Pure and Applied Mathematics, 2023 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Basım Tarihi: 2023
  • Doi Numarası: 10.1007/s13226-023-00459-2
  • Dergi Adı: Indian Journal of Pure and Applied Mathematics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, INSPEC, zbMATH
  • Anahtar Kelimeler: Convolution sums, Cusp forms, Dedekind eta function, Eisenstein series, Modular forms, Quadratic forms, Representations, Sum of divisors function
  • Recep Tayyip Erdoğan Üniversitesi Adresli: Evet

Özet

For a positive integer n we evaluate the convolution sum ∑al+bm=nσ(l)σ3(m) for (a, b) = (1 , 7) , (7, 1), (1, 8), (8, 1), (1, 9) and (9, 1). We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of n by the forms x12+x22+x32+x42+2(x52+x62+x72+x82+x92+x102+x112+x122),x12+x22+x32+x42+x52+x62+x72+x82+2(x92+x102+x112+x122),x12+x1x2+x22+x32+x3x4+x42+3(x52+x5x6+x62+x72+x7x8+x82+x92+x9x10+x102+x112+x11x12+x122) , x12+x1x2+x22+x32+x3x4+x42+x52+x5x6+x62+x72+x7x8+x82+3(x92+x9x10+x102+x112+x11x12+x122) . We use a modular form approach.