  In signal processing the Rudin-Shapiro polynomials have good autocorrelation
properties and their values on the unit circle are small. Binary sequences with
low autocorrelation coefficients are of interest in radar, sonar, and
communication systems. In this paper we study the oscillation of the modulus of
the Rudin-Shapiro polynomials on the unit circle. We also show that the
Rudin-Shapiro polynomials $P_k$ and $Q_k$ of degree $n-1$ with $n := 2^k$ have
$o(n)$ zeros on the unit circle. This should be compared with a result of B.
Conrey, A. Granville, B. Poonen, and K. Soundararajan stating that for odd
primes $p$ the Fekete polynomials $f_p$ of degree $p-1$ have asymptotically
$\kappa_0 p$ zeros on the unit circle, where $0.500813>\kappa_0>0.500668$. Our
approach is based heavily on the Saffari and Montgomery conjectures proved
recently by B. Rodgers. We also prove that there are absolute constants $c_1 >
0$ and $c_2 > 0$ such that the $k$-th Rudin-Shapiro polynomials $P_k$ and $Q_k$
of degree $n-1 = 2^k-1$ have at least $c_2n$ zeros in the annulus $$\left \{z
\in {\Bbb C}: 1 - \frac{c_1}{n} < |z| < 1 + \frac{c_1}{n} \right \}\,.$$
