# Evaluation of Convolution Sums entailing mixed Divisor Functions for a Class of Levels

### New Zealand Journal of Mathematics

Vol. 50, (2020), Pages 125-180

Ebenezer Ntienjem

Centre for Research in Algebra and Number Theory School of Mathematics and Statistics Carleton University 1125 Colonel By Drive Ottawa, Ontario, K1S 5B6, Canada

Abstract Let $0< n,\alpha,\beta\in\mathbb{N}$ be such that gcd(α,β) = 1. We carry out the evaluation of the convolution sums **** and **** for all levels $\alpha\beta\in\mathbb{N}$, by using in particular modular forms. We next apply convolution sums belonging to this class of levels to determine formulae for the number of representations of a positive integer n by the quadratic forms in twelve variables **** when the level $\alpha\beta\equiv 0\pmod{4}$, and **** when the level $\alpha\beta\equiv 0\pmod{3}$. Our approach is then illustrated by explicitly evaluating the convolution sum for αβ = 3, 6, 7, 8, 9, 12, 14, 15, 16, 18, 20, 21, 27, 32. These convolution sums are then applied to determine explicit formulae for the number of representations of a positive integer n by quadratic forms in twelve variables.

Keywords Sums of Divisors; Dedekind eta function; Convolution Sums; Modular Forms; Dirichlet Characters; Eisenstein forms; Cusp Forms; Octonary quadratic Forms; Number of Representations.

Classification (MSC2000) 11A25, 11F11, 11F20, 11E20, 11E25, 11F27.