Turán’s pure power sum problem
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- by A. Y. Cheer and D. A. Goldston PDF
- Math. Comp. 65 (1996), 1349-1358 Request permission
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 $n\to \infty$.References
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Additional Information
- A. Y. Cheer
- Affiliation: Department of Mathematics and Computer Science, San Jose State University, San Jose, California 95192
- Email: aycheer@ucdavis.edu
- D. A. Goldston
- Affiliation: Department of Mathematics and Institute of Theoretical Dynamics, University of California, Davis, California 95616
- MR Author ID: 74830
- ORCID: 0000-0002-6319-2367
- Email: goldston@jupiter.sjsu.edu
- 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. Research of the second author was supported in part by NSF Grant DMS9205533 and NSF Computing Research Environments Award 9303986
- © Copyright 1996 American Mathematical Society
- Journal: Math. Comp. 65 (1996), 1349-1358
- MSC (1991): Primary 11N30
- DOI: https://doi.org/10.1090/S0025-5718-96-00744-2
- MathSciNet review: 1348041