Cyclicity and iterated logarithms in the Dirichlet space

Let $D(\mu )$ denote a harmonically weighted Dirichlet space on the unit disc $\mathbb{D}$. We show that outer functions $f\in D(\mu )$ are cyclic in $D(\mu )$, whenever $\log f$ belongs to the Pick-Smirnov class $N^{+}(D(\mu ))$. If $f$ has $H^{\mathrm{\infty }}$-norm less than or equal to 1, then cyclicity can also be checked via iterated logarithms. For example, we show that such outer functions $f$ are cyclic, whenever $\log (1+\log (1\mathrm{/}f))\in N^{+}(D(\mu ))$. This condition can be checked by verifying that $\log (1+\log (1\mathrm{/}f))\in D(\mu )$. If $f$ satisfies a mild extra condition, then the conditions also become necessary for cyclicity.