  Let $\xi_0,\xi_1,\ldots$ be i.i.d. random variables with zero mean and unit
variance. Consider a random Taylor series of the form $f(z)=\sum_{k=0}^\infty
\xi_k c_k z^k$, where $c_0,c_1,\ldots$ is a real sequence such that $c_n^2$ is
regularly varying with index $\gamma-1$, where $\gamma>0$. We prove that
$\mathbb{E} N[0,1-\epsilon] \sim \frac{\sqrt{\gamma}}{2\pi} |\log \epsilon|$ as
$\epsilon \downarrow 0$, where $N[0,r]$ denotes the number of real zeroes of
$f$ in the interval $[0,r]$.
