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Merit Factors of Polynomials Formed by Jacobi Symbols

Published online by Cambridge University Press:  20 November 2018

Peter Borwein  and
Kwok-Kwong Stephen Choi
Peter Borwein
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC, V5A 1S6
Kwok-Kwong Stephen Choi
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
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Abstract

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We give explicit formulas for the ${{L}_{4}}$ norm (or equivalently for the merit factors) of various sequences of polynomials related to the polynomials

$$f\left( z \right):=\,\sum\limits_{n=0}^{N-1}{\left( \frac{n}{N} \right){{z}^{n}}.}$$

and

$${{f}_{t}}(z)\,=\,\sum\limits_{n=0}^{N-1}{\left( \frac{n+t}{N} \right){{z}^{n}}.}$$

where $\left( \frac{.}{N} \right)$ is the Jacobi symbol.

Two cases of particular interest are when $N\,=\,pq$ is a product of two primes and $p\,=\,q\,+\,2$ or $p\,=\,q\,+\,4$. This extends work of Høholdt, Jensen and Jensen and of the authors.

This study arises from a number of conjectures of Erdős, Littlewood and others that concern the norms of polynomials with −1, 1 coefficients on the disc. The current best examples are of the above form when $N$ is prime and it is natural to see what happens for composite $N$.

Keywords

11J5411B8312-04Character polynomialClass Number−1, 1 coefficientsMerit factorFekete polynomialsTuryn PolynomialsLittlewood polynomialsTwin PrimesJacobi Symbols
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 1 , 01 February 2001 , pp. 33 - 50
Copyright
Copyright © Canadian Mathematical Society 2001

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