An extremal property of Fekete polynomials
Authors:
Peter Borwein, KwokKwong Stephen Choi and Soroosh Yazdani
Journal:
Proc. Amer. Math. Soc. 129 (2001), 1927
MSC (2000):
Primary 11J54, 11B83
Published electronically:
July 21, 2000
MathSciNet review:
1784013
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: The Fekete polynomials are defined as
where is the Legendre symbol. These polynomials arise in a number of contexts in analysis and number theory. For example, after cyclic permutation they provide sequences with smallest known norm out of the polynomials with coefficients. The main purpose of this paper is to prove the following extremal property that characterizes the Fekete polynomials by their size at roots of unity. Theorem 0.1. Let with odd and . If
then must be an odd prime and is . Here This result also gives a partial answer to a problem of Harvey Cohn on character sums.
 1.
Tom
M. Apostol, Introduction to analytic number theory,
SpringerVerlag, New YorkHeidelberg, 1976. Undergraduate Texts in
Mathematics. MR
0434929 (55 #7892)
 2.
Leonard
D. Baumert, Cyclic difference sets, Lecture Notes in
Mathematics, Vol. 182, SpringerVerlag, BerlinNew York, 1971. MR 0282863
(44 #97)
 3.
A. Biró, Notes on a Problem of H. Cohn, J. Number Theory, 77 (1999), 200208. CMP 99:16
 4.
P. Borwein and KK. Choi, Explicit Merit Factor Formulae For Fekete and Turyn Polynomials, (in press).
 5.
KK. Choi and MK Siu, CounterExamples to a Problem of Cohn on Classifying Characters, J. Number Theory, to appear.
 6.
B. Conrey, A.Granville and B.Poonen, Zeros of Fekete Polynomials, (in press).
 7.
SL Ma, MK Siu and Z Zheng, On a Problem of Cohn on Character Sums, (in press).
 8.
H.L. Montgomery, An Exponential Sum Formed with the Legendre Symbol, Acta Arith, 37 (1980), 375380
 9.
Hugh
L. Montgomery, Ten lectures on the interface between analytic
number theory and harmonic analysis, CBMS Regional Conference Series
in Mathematics, vol. 84, Published for the Conference Board of the
Mathematical Sciences, Washington, DC; by the American Mathematical
Society, Providence, RI, 1994. MR 1297543
(96i:11002)
 1.
 T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, Berlin, 1976. MR 55:7892
 2.
 L. D. Baumert, Cyclic Difference Sets, LNM 182, SpringerVerlag, Berlin, 1971. MR 44:97
 3.
 A. Biró, Notes on a Problem of H. Cohn, J. Number Theory, 77 (1999), 200208. CMP 99:16
 4.
 P. Borwein and KK. Choi, Explicit Merit Factor Formulae For Fekete and Turyn Polynomials, (in press).
 5.
 KK. Choi and MK Siu, CounterExamples to a Problem of Cohn on Classifying Characters, J. Number Theory, to appear.
 6.
 B. Conrey, A.Granville and B.Poonen, Zeros of Fekete Polynomials, (in press).
 7.
 SL Ma, MK Siu and Z Zheng, On a Problem of Cohn on Character Sums, (in press).
 8.
 H.L. Montgomery, An Exponential Sum Formed with the Legendre Symbol, Acta Arith, 37 (1980), 375380
 9.
 H.L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS, No 84, AMS, 1994. MR 96i:11002
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Additional Information
Peter Borwein
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email:
pborwein@math.sfu.ca
KwokKwong Stephen Choi
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, SAR, China
Email:
choi@maths.hku.hk
Soroosh Yazdani
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email:
syazdani@undergrad.math.uwaterloo.ca
DOI:
http://dx.doi.org/10.1090/S0002993900057981
PII:
S 00029939(00)057981
Keywords:
Class number,
$\pm1$ coefficients,
merit factor,
Fekete polynomials,
Turyn polynomials,
Littlewood polynomials
Received by editor(s):
March 15, 1999
Published electronically:
July 21, 2000
Additional Notes:
The research of P. Borwein is supported, in part, by NSERC of Canada. K.K. Choi is a Pacific Institute of Mathematics Postdoctoral Fellow and the Institute’s support is gratefully acknowledged.
Communicated by:
Dennis A. Hejhal
Article copyright:
© Copyright 2000
Copyright held by the authors
