An extremal property of Fekete polynomials

Authors:
Peter Borwein, Kwok-Kwong Stephen Choi and Soroosh Yazdani

Journal:
Proc. Amer. Math. Soc. **129** (2001), 19-27

MSC (2000):
Primary 11J54, 11B83

DOI:
https://doi.org/10.1090/S0002-9939-00-05798-1

Published electronically:
July 21, 2000

MathSciNet review:
1784013

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Abstract | References | Similar Articles | Additional Information

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.**T. M. Apostol,*Introduction to Analytic Number Theory*, Springer-Verlag, Berlin, 1976. MR**55:7892****2.**L. D. Baumert,*Cyclic Difference Sets*, LNM**182**, Springer-Verlag, Berlin, 1971. MR**44:97****3.**A. Biró,*Notes on a Problem of H. Cohn*, J. Number Theory,**77**(1999), 200-208. CMP**99:16****4.**P. Borwein and K-K. Choi,*Explicit Merit Factor Formulae For Fekete and Turyn Polynomials*, (in press).**5.**K-K. Choi and M-K Siu,*Counter-Examples 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.**S-L Ma, M-K 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), 375-380**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

**Kwok-Kwong 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:
https://doi.org/10.1090/S0002-9939-00-05798-1

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