  In this paper we investigate the asymptotic behavior of the Riesz $s$-energy
of the first $N$ points of a greedy $s$-energy sequence on the unit circle, for
all values of $s$ in the range $0\leq s<\infty$ (identifying as usual the case
$s=0$ with the logarithmic energy). In the context of the unit circle, greedy
$s$-energy sequences coincide with the classical Leja sequences constructed
using the logarithmic potential. We obtain first-order and second-order
asymptotic results. The key idea is to express the Riesz $s$-energy of the
first $N$ points of a greedy $s$-energy sequence in terms of the binary
representation of $N$. Motivated by our results, we pose some conjectures for
general sequences on the unit circle.
