Explicit merit factor formulae for Fekete and Turyn polynomials

Abstract:

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 $\left (\frac {\cdot }{q}\right )$ 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 $[\cdot ]$ denotes the nearest integer, satisfies \[ \|R_{q}\|_{4}^{4} = \frac {7q^{2}}{6}- {q} - \frac {1}{6} - \gamma _{q} \] where \[ \gamma _{q}: = \begin {cases} h(-q) (h(-q)-4) & \text {if $q \equiv 1,5 \pmod 8$},\\ 12 (h(-q))^{2} & \text {if $q \equiv 3 \pmod 8$}, \\ 0 & \text {if $q \equiv 7 \pmod 8$}. \end {cases} \] 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} \left (\frac {-1}{q}\right ) \right ) \left |{\displaystyle \sum _{n=1}^{q-1}n \left (\frac {n+t}{q}\right )} \right |^{2}, \end{align*} and $\| f_{q}^{q-t+1} \|_{4}^{4}= \| f_{q}^{t} \|_{4}^{4}$ if $1 \le t \le (q+1)/2$.