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Mathematics of Computation

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Turán's Pure Power Sum Problem

Authors: A. Y. Cheer and D. A. Goldston
Journal: Math. Comp. 65 (1996), 1349-1358
MSC (1991): Primary 11N30
MathSciNet review: 1348041
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Abstract: 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 $.

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Additional Information

A. Y. Cheer
Affiliation: Department of Mathematics and Computer Science, San Jose State University, San Jose, California 95192

D. A. Goldston
Affiliation: Department of Mathematics and Institute of Theoretical Dynamics, University of California, Davis, California 95616

Keywords: Tur{\'{a}}n's method
Received by editor(s): March 4, 1995
Additional Notes: Research of the first author was supported in part by the Institute for Theoretical Dynamics, University of California at Davis. \endgraf Research of the second author was supported in part by NSF Grant DMS9205533 and NSF Computing Research Environments Award 9303986
Article copyright: © Copyright 1996 American Mathematical Society