A $k$-proper edge-coloring of a graph G is called adjacent vertex-distinguishing if any two adjacent vertices are distinguished by the set of colors appearing in the edges incident to each vertex. The smallest value $k$ for which $G$ admits such coloring is denoted by $χ'_a(G)$. We prove that $χ'_a(G) = 2R + 1$ for most circulant graphs $C_n([1, R])$.