Turán's Pure Power Sum Problem

Let $1 = z_{1} \ge |z_{2}|\ge \cdots \ge |z_{n}|$ be $n$ complex numbers, and consider the power sums $s_{\nu }= {z_{1}}^{\nu }+ {z_{2}}^{\nu }+ \cdots + {z_{n}}^{\nu }$, $1\le \nu \le n$. Put $R_{n} = \min \max _{1\le \nu \le n} |s_{\nu }|$, where the minimum is over all possible complex numbers satisfying the above. Turán conjectured that $R_{n} > A$, for $A$ some positive absolute constant. Atkinson proved this conjecture by showing $R_{n} > 1/6$. It is now known that $1/2<R_{n} < 1$, for $n\ge 2$. Determining whether $R_{n} \to 1$ or approaches some other limiting value as $n\to \infty$ is still an open problem. Our calculations show that an upper bound for $R_{n}$ decreases for $n\le 55$, suggesting that $R_{n}$ decreases to a limiting value less than $0.7$ as $n\to \infty$.