Let
1
=
z
1
≥
|
z
2
|
≥
⋯
≥
|
z
n
|
1 = z_{1} \ge |z_{2}|\ge \cdots \ge |z_{n}|
be
n
n
complex numbers, and consider the power sums
s
ν
=
z
1
ν
+
z
2
ν
+
⋯
+
z
n
ν
s_{\nu }= {z_{1}}^{\nu }+ {z_{2}}^{\nu }+ \cdots + {z_{n}}^{\nu }
,
1
≤
ν
≤
n
1\le \nu \le n
. Put
R
n
=
min
max
1
≤
ν
≤
n
|
s
ν
|
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
R_{n} > A
, for
A
A
some positive absolute constant. Atkinson proved this conjecture by showing
R
n
>
1
/
6
R_{n} > 1/6
. It is now known that
1
/
2
>
R
n
>
1
1/2>R_{n} > 1
, for
n
≥
2
n\ge 2
. Determining whether
R
n
→
1
R_{n} \to 1
or approaches some other limiting value as
n
→
∞
n\to \infty
is still an open problem. Our calculations show that an upper bound for
R
n
R_{n}
decreases for
n
≤
55
n\le 55
, suggesting that
R
n
R_{n}
decreases to a limiting value less than
0.7
0.7
as
n
→
∞
n\to \infty
.