Using a generalization due to Lerch [Bull. Int. Acad. François Joseph 3 (1896)] of a classical lemma of Zolotarev, employed in Zolotarev's proof of the law of quadratic reciprocity, we determine necessary and sufficient conditions for the difference of two Dedekind sums to be in 8ℤ. These yield new necessary conditions for equality of two Dedekind sums. In addition, we resolve a conjecture of Girstmair [arXiv:1501.00655].

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...