Let \(H^2_d\) be the Drury–Arveson space, and let \(f\in H^2_d\) have bounded argument and no zeros in \(\mathbb B_d\) . We show that f is cyclic in \(H^2_d\) if and only if \(\log f\) belongs to the Pick-Smirnov class \(N^+(H^2_d)\) . Furthermore, for non-vanishing functions \(f\in H^2_d\) with bounded argument and \(H^\infty \) -norm less than 1, cyclicity can also be tested via iterated logarithms. For example, we show that f is cyclic if and only if \(\log (1+\log (1/f))\in N^+(H^2_d)\) . Thus, a sufficient condition for cyclicity is that \(\log (1+\log (1/f))\in H^2_d\) . More generally, our results hold for all radially weighted Besov spaces that also are complete Pick spaces.