Let 1 = z 1 ≥ |z 2 | ≥ · · · ≥ |zn| be n complex numbers, and consider the power sums sν = z 1 ν + z 2 ν + · · · + zn ν , 1 ≤ ν ≤ n. Put Rn = min max 1≤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

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... 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 → ∞.