Elementary Evaluation of Convolution Sums Involving the Divisor Function for a Class of Levels

Abstract

We discuss the evaluation of convolution sums involving the divisor function, \(\underset{\substack{ {(l,m)\in\mathbb{N}^{2}} \\ {\alpha\,l+\beta\,m=n} } }{\sum}\sigma(l)\sigma(m)\), for the class of levels \(\alpha\beta\) belonging to all natural numbers. The evaluation of convolution sums belonging to this class is achieved by applying modular forms and primitive Dirichlet characters. We illustrate our method with the explicit examples for the levels \(\alpha\beta=33\), \(40\), \(45\), \(50\), \(54\), and \(56\). As a corollary, the known convolution sums for the levels \(\alpha\beta=10\), \(11\), \(12\), \(15\), \(16\), \(18\), \(24\), \(25\), \(27\), \(32\) and \(36\) are improved when we revisit their evaluations. If the level \(\alpha\beta\equiv 0 \pmod{4}\), we determine natural numbers \(a,b\) and use the evaluated convolution sums together with other known convolution sums to carry out the number of representations of \(n\) by the octonary quadratic forms \(a\,\underset{i=1}{\overset{4}{\sum}}x_{i}^{2}+ b\,\underset{i=5}{\overset{8}{\sum}}x_{i}^{2}\). Similarly, if the level \(\alpha\beta\equiv 0 \pmod{3}\), we compute natural numbers \(c,d\) and make use of the evaluated convolution sums together with other known convolution sums to determine the number of representations of \(n\) by the octonary quadratic forms \(c\,\underset{i=1}{\overset{2}{\sum}}\,(\,x_{2i-1}^{2}+ x_{2i-1}x_{2i} + x_{2i}^{2}\,) + d\,\underset{i=3}{\overset{4}{\sum}}\,(\,x_{2i-1}^{2}+ x_{2i-1}x_{2i} + x_{2i}^{2}\,)\). In addition, we determine formulae for the number of representations of a positive integer \(n\) when \((a,b)=(1,1)\), \((1,3)\), \((1,6)\), \((2,3)\).