We give explicit formulas for the
L
4
L_{4}
norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials
\[
f
q
(
z
)
:=
∑
k
=
1
q
−
1
(
k
q
)
z
k
f_{q}(z) := \sum ^{q-1}_{k=1} \left (\frac {k}{q}\right ) z^{k}
\]
where
(
⋅
q
)
\left (\frac {\cdot }{q}\right )
is the Legendre symbol. For example for
q
q
an odd prime,
\[
‖
f
q
‖
4
4
:=
5
q
2
3
−
3
q
+
4
3
−
12
(
h
(
−
q
)
)
2
\|f_{q}\|_{4}^{4} : = \frac {5q^{2}}{3}-3q+ \frac {4}{3} - 12 (h(-q))^{2}
\]
where
h
(
−
q
)
h(-q)
is the class number of
Q
(
−
q
)
\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
f_{q}
. This is the sequence that has the largest known asymptotic merit factor. Explicitly,
\[
R
q
(
z
)
:=
∑
k
=
0
q
−
1
(
k
+
[
q
/
4
]
q
)
z
k
R_{q}(z) := \sum ^{q-1}_{k=0} \left (\frac {k+[q/4] }{q}\right ) z^{k}
\]
where
[
⋅
]
[\cdot ]
denotes the nearest integer, satisfies
\[
‖
R
q
‖
4
4
=
7
q
2
6
−
q
−
1
6
−
γ
q
\|R_{q}\|_{4}^{4} = \frac {7q^{2}}{6}- {q} - \frac {1}{6} - \gamma _{q}
\]
where
\[
γ
q
:=
{
h
(
−
q
)
(
h
(
−
q
)
−
4
)
a
m
p
;
if
q
≡
1
,
5
(
mod
8
)
,
12
(
h
(
−
q
)
)
2
a
m
p
;
if
q
≡
3
(
mod
8
)
,
0
a
m
p
;
if
q
≡
7
(
mod
8
)
.
\gamma _{q}: = \begin {cases} h(-q) (h(-q)-4) & \text {if $q \equiv 1,5 \pmod 8$},\\ 12 (h(-q))^{2} & \text {if $q \equiv 3 \pmod 8$}, \\ 0 & \text {if $q \equiv 7 \pmod 8$}. \end {cases}
\]
Indeed we derive a closed form for the
L
4
L_{4}
norm of all shifted Fekete polynomials
\[
f
q
t
(
z
)
:=
∑
k
=
0
q
−
1
(
k
+
t
q
)
z
k
.
f_{q}^{t}(z) := \sum ^{q-1}_{k=0} \left (\frac {k+t}{q}\right ) z^{k}.
\]
Namely
‖
f
q
t
‖
4
4
a
m
p
;
=
1
3
(
5
q
2
+
3
q
+
4
)
+
8
t
2
−
4
q
t
−
8
t
a
m
p
;
−
8
q
2
(
1
−
1
2
(
−
1
q
)
)
|
∑
n
=
1
q
−
1
n
(
n
+
t
q
)
|
2
,
\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
\| f_{q}^{q-t+1} \|_{4}^{4}= \| f_{q}^{t} \|_{4}^{4}
if
1
≤
t
≤
(
q
+
1
)
/
2
1 \le t \le (q+1)/2
.