Explicit merit factor formulae for Fekete and Turyn polynomials

• Published in: Transactions of the American Mathematical Society 2002-01, Vol.354 (1), p.219-234
• Main Authors: Borwein, Peter, Choi, Kwok-Kwong Stephen
• Format: Article
• Language: English
• Subjects: Algebra; Coefficients; Exact sciences and technology; Field theory and polynomials; Integers; Mathematical sequences; Mathematical theorems; Mathematics; Number theory; Polynomials; Prime numbers; Sciences and techniques of general use
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-01-02859-8

We give explicit formulas for the L_{4} norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials \[ f_{q}(z) := \sum ^{q-1}_{k=1} \left (\frac{k}{q}\right ) z^{k} \] where \left (\frac{·}{q}\right ) is the Legendre symbol. For example for q an odd prime, \[ \|f_{q}\|_{4}^{4} : = \frac{5q^{2}}{3}-3q+ \frac{4}{3} - 12 (h(-q))^{2} \] where h(-q) is the class number of \mathbb{Q}(\sqrt {-q}). Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of f_{q}. This is the sequence that has the largest known asymptotic merit factor. Explicitly, \[ R_{q}(z) := \sum ^{q-1}_{k=0} \left (\frac{k+[q/4] }{q}\right ) z^{k} \] where [·] denotes the nearest integer, satisfies \[ \|R_{q}\|_{4}^{4} = \frac{7q^{2}}{6}- {q} - \frac{1}{6} - \gamma _{q} \] where \[ \gamma _{q}: = \begin{cases} h(-q) (h(-q)-4) & \text{if} \quad q \equiv 1,5 \pmod 8, 12 (h(-q))^{2} & \text{if} \quad q \equiv 3 \pmod 8, 0 & \text{if} \quad q \equiv 7 \pmod 8. \end{cases} \] Indeed we derive a closed form for the L_{4} norm of all shifted Fekete polynomials \[ f_{q}^{t}(z) := \sum ^{q-1}_{k=0} \left (\frac{k+t}{q}\right ) z^{k}. \] Namely \begin{align*} | f_{q}^{t} \|_{4}^{4} &= \frac{1}{3}(5q^{2}+3q+4)+8t^{2}-4qt-8t &\quad -\frac{8}{q^{2}}\left ( 1-\frac{1}{2}\leg {-1}{q} \right ) \left |{\displaystyle \sum _{n=1}^{q-1}n\leg {n+t}{q}}\right |^{2}, \end{align*} and | f_{q}^{q-t+1} \|_{4}^{4}= | f_{q}^{t} \|_{4}^{4} if 1 \le t \le (q+1)/2.