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For the Riesz and logarithmic potentials, we consider greedy energy sequences$(a_n)_{n=0}^\infty$ on the unit circle $S^1$, constructed in such a way thatfor every $n\geq 1$, the discrete potential generated by the first $n$ points$a_0,\ldots,a_{n-1}$ of the sequence attains its minimum (say $U_n$) at $a_n$.We obtain asymptotic formulae that describe the behavior of $U_n$ as$n\to\infty$, in terms of certain bounded arithmetic functions with a doublingperiodicity property. As previously shown in \cite{LopMc2}, after properlytranslating and scaling $U_n$, one obtains a new sequence $(F_n)$ that isbounded and divergent. We find the exact value of $\liminf F_n$ (the value of$\limsup F_n$ was already given in \cite{LopMc2}), and show that the interval$[\liminf F_n,\limsup F_n]$ comprises all the limit points of the sequence$(F_n)$.

Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle