Turán's pure power sum problem

Abstract. Let 1 = z1 ≥ |z2| ≥ · · · ≥ |zn| be n complex numbers, and consider the power sums sν = z1 + z2 + · · · + zn , 1 ≤ ν ≤ n. Put Rn = min max1≤ν≤n |sν |, where the minimum is over all possible complex numbers satisfying the above. Turán conjectured that Rn > A, for A some positive absolute constant. Atkinson proved this conjecture by showing Rn > 1/6. It is now known that 1/2 < Rn < 1, for n ≥ 2. Determining whether Rn → 1 or approaches some other limiting value as n→∞ is still an open problem. Our calculations show that an upper bound for Rn decreases for n ≤ 55, suggesting that Rn decreases to a limiting value less than 0.7 as n→∞.