Evaluation of the convolution sum $$\displaystyle \varvec{\sum _{al+bm=n} \sigma (l)\sigma _ 3 (m) }$$ for $$\varvec{(a,b) = (1,7), (7,1), (1,8), (8,1), (1,9), (9,1)}$$ and representations by certain quadratic forms in twelve variables

Abstract

Evaluation of the convolution sum $$\displaystyle \varvec{\sum _{al+bm=n} \sigma (l)\sigma _ 3 (m) }$$ for $$\varvec{(a,b) = (1,7), (7,1), (1,8), (8,1), (1,9), (9,1)}$$ and representations by certain quadratic forms in twelve variables