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 | References | Similar Articles | Additional Information

Abstract: Let be complex numbers, and consider the power sums , . Put , where the minimum is over all possible complex numbers satisfying the above. Turán conjectured that , for some positive absolute constant. Atkinson proved this conjecture by showing . It is now known that , for . Determining whether or approaches some other limiting value as is still an open problem. Our calculations show that an upper bound for decreases for , suggesting that decreases to a limiting value less than as .

**1.**F. V. Atkinson,*On sums of powers of complex numbers*, Acta Math. Acad. Sci. Hungar.**12**(1961), 185–188 (English, with Russian summary). MR**0126418****2.**F. V. Atkinson,*Some further estimates concerning sums of powers of complex numbers.*, Acta Math. Acad. Sci. Hungar**20**(1969), 193–210. MR**0239084****3.**A. Biró,*On a problem of Turán concerning sums of powers of complex numbers*, Acta Math. Hungar.**65**(1994), no. 3, 209–216. MR**1281430**, 10.1007/BF01875148**4.**János Komlós, András Sárközi, and Endre Szemerédi,*On sums of powers of complex numbers*, Mat. Lapok**15**(1964), 337–347 (Hungarian, with Russian and English summaries). MR**0202674****5.**Julian Ławrynowicz,*Remark on a problem of P. Turán*, Bull. Soc. Sci. Lettres Lódź**11**(1960), no. 1, 4. MR**0126664****6.**Julian Ławrynowicz,*Calculation of a minimum maximorum of complex numbers*, Bull. Soc. Sci. Lettres Lódź**11**(1960), no. 2, 9. MR**0126663****7.**J. Ławrynowicz,*Remark on power-sums of complex numbers*, Acta Math. Acad. Sci. Hungar.**18**(1967), 279–281. MR**0217288****8.**Paul Turán,*On a new method of analysis and its applications*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. With the assistance of G. Halász and J. Pintz; With a foreword by Vera T. Sós; A Wiley-Interscience Publication. MR**749389****9.**Paul Turán,*Collected papers of Paul Turán. Vol. 1–3*, Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1990. Edited by Paul Erdős. MR**1056293**

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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

Email:
goldston@jupiter.sjsu.edu

DOI:
https://doi.org/10.1090/S0025-5718-96-00744-2

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