Let A = (a0 , a1 , · · · , an−1 ) be a finite sequence of complex numbers with modulus 1 of length
n. Let
A(z) := a0 + a1 z + · · · + an−1 z n−1
be the unimodular polynomial associated with the sequence A and z := e2πi/n . In this talk, we give
an exact formula of the L4 norm for A over the unit circle, namely, if n is an odd positive integer,
then
n−1
n−1
1X
4 X
4
a 4
kAk4 =
|A(ζ )| − 3
|A(ζ a )|2 < A(ζ a )A2 (ζ a )
n a=0
n a=0
n−1
n−1
2
4 X 8 X
a 2
a
− 3
< A(ζ ) A1 (ζ ) + 3
|A(ζ a )A1 (ζ a )|2 ,
n a=0
n a=0
where
A1 (z) :=
n−1
X
`=0
`a` z
`
and
A2 (z) :=
n−1
X
`2 a` z ` .
`=0
Using this formula, we are able to prove that if an−` = εa` for all 1 ≤ ` < n for some fixed
complex number ε with |ε| = 1 and a0 := ε−1/2 , then we have
4
5
kAk44 ≥ n2 − 2n + .
3
3
P a a
The main term of lower bound is the best possible and is attained by A(z) := 1 + p−1
a=1 p z ,
where p· is the Legendre symbol and p is a prime ≡ 1 (mod 4). As a corollary, we can show
that if A(z) is a reciprocal polynomial of even degree n − 1, then kAk44 ≥ 53 n2 + O(n3/2 ). Also,
our result shows that the largest asymptotic merit factor for reciprocal Littlewood polynomials of
even degree is 3/2.