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# Merit Factors of Polynomials Formed by Jacobi Symbols

Published:2001-02-01
Printed: Feb 2001
• Peter Borwein
• Kwok-Kwong Stephen Choi
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## Abstract

We give explicit formulas for the $L_4$ norm (or equivalently for the merit factors) of various sequences of polynomials related to the polynomials $$f(z) := \sum_{n=0}^{N-1} \leg{n}{N} z^n.$$ and $$f_t(z) = \sum_{n=0}^{N-1} \leg{n+t}{N} z^n.$$ where $(\frac{\cdot}{N})$ 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 H{\o}holdt, Jensen and Jensen and of the authors. This study arises from a number of conjectures of Erd\H{o}s, 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: Character polynomial, Class Number, $-1, 1$ coefficients, Merit factor, Fekete polynomials, Turyn Polynomials, Littlewood polynomials, Twin Primes, Jacobi Symbols
 MSC Classifications: 11J54 - Small fractional parts of polynomials and generalizations 11B83 - Special sequences and polynomials 12-04 - Explicit machine computation and programs (not the theory of computation or programming)