We investigate the numbers of complex zeros of Littlewood polynomials
$p\left( z \right)$
(polynomials
with coefficients {−1, 1}) inside or on the unit circle
$\left| z \right|\,=\,1$
, denoted by
$N\left( p \right)$
and
$U\left( p \right)$
, respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in
the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain
explicit formulas for
$N\left( p \right)$
,
$U\left( p \right)$
for polynomials
$p\left( z \right)$
of these types. We show that if
$n\,+\,1$
is a prime number, then for each integer
$k,\,0\,⩽\,k\,⩽\,n-1$
, there exists a Littlewood polynomial
$p\left( z \right)$
of degree
$n$
with
$N\left( p \right)\,=\,k$
and
$U\left( p \right)\,=\,0$
. Furthermore, we describe some cases where the ratios
$N\left( p \right)/n$
and
$U\left( p \right)/n$
have limits as
$n\,\to \,\infty $
and find the corresponding limit values.