The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $\begin{array}{c}\sum _{\genfrac{}{}{0pt}{}{(l,m)\in {\mathbb{N}}_0^2}{\alpha l+\beta m=n}}\sigma \left(l\right)\sigma \left(m\right),\end{array}$ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by...

We use hyperbolic geometry to study the limiting behavior of the average number of ways of expressing a number as the sum of two coprime squares. An alternative viewpoint using analytic number theory is also given.