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Turán's Pure Power Sum Problem

Authors:	A Y Cheer D A Goldston

Institution:	Department of Mathematics and Computer Science, San Jose State University, San Jose, California 95192 ; Department of Mathematics and Institute of Theoretical Dynamics, University of California, Davis, California 95616

Abstract:	Let $1 = z_{1} \ge \|z_{2}\|\ge \cdots \ge \|z_{n}\|$ be 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 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 as $n\to \infty$ .

Keywords:	Tur{\'{a}}n's method

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