Merit Factors of Polynomials Formed by Jacobi Symbols

Borwein, Peter ; Choi, Kwok-Kwong Stephen

Canadian journal of mathematics, Tome 53 (2001) no. 1, pp. 33-50

Résumé

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 $N$ .

Détail

DOI : 10.4153/CJM-2001-002-6

Mots-clés : 11J54, 11B83, 12-04, Character polynomial, Class Number, −1, 1 coefficients, Merit factor, Fekete polynomials, Turyn Polynomials, Littlewood polynomials, Twin Primes, Jacobi Symbols

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