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Merit Factors of Polynomials Formed by Jacobi Symbols


Published:20010201
Printed: Feb 2001
Peter Borwein
KwokKwong Stephen Choi
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
$$
f(z) := \sum_{n=0}^{N1} \leg{n}{N} z^n.
$$
and
$$
f_t(z) = \sum_{n=0}^{N1} \leg{n+t}{N} z^n.
$$
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
Character polynomial, Class Number, $1, 1$ coefficients, Merit factor, Fekete polynomials, Turyn Polynomials, Littlewood polynomials, Twin Primes, Jacobi Symbols
