# Evaluation of the Convolution Sums

## Creator:

Ntienjem, Ebenezer

2015

## Abstract:

For all positive integers $n$ we evaluate the convolution sums $\displaystyle\overset{}{\underset{ \begin{array}{c} {(l,m) \in \mathbb{N}^2} \\ {\alpha l+\beta m=n} \end{array} } {\sum}}\sigma(l)\sigma(m)$, where $(\alpha,\beta) = (1,14), (2,7), (1,26), (2,13), (1,28)$, $(4,7), (1,30), (2,15), (3,10), (5,6)$. Using some of the evaluations of these convolution sums we determine formulae for the number of representations of $n$ by the octonary quadratic forms \begin{equation*} x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}+ 7 (x_{5}^{2} + x_{6}^{2} + x_{7}^{2} + x_{8}^{2}) \end{equation*} and \begin{equation*} a(\,x_{1}^{2} + x_{2}^{2} + x_{1}x_{2} + x_{3}^{2} + x_{3}x_{4} + x_{4}^{2}\,) + b (\,x_{5}^{2} + x_{5}x_{6} + x_{6}^{2} + x_{7}^{2} + x_{7}x_{8} + x_{8}^{2}\,), \end{equation*} where $(a,b)$ stands for $(1, 10)$ or $(2,5)$.

Mathematics

English

## Publisher:

Carleton University

## Thesis Degree Name:

Master of Science:
M.Sc.

Master's

Pure Mathematics

## Parent Collection:

Theses and Dissertations