# On the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle

## TL;DR

This paper investigates the oscillation and zero distribution of Rudin-Shapiro polynomials on the unit circle, revealing they have fewer zeros on the circle than Fekete polynomials and many zeros near it.

## Contribution

It provides new bounds on the number of zeros of Rudin-Shapiro polynomials on and near the unit circle, extending understanding of their oscillatory behavior.

## Key findings

- Rudin-Shapiro polynomials have o(n) zeros on the unit circle.
- At least c2n zeros lie within a narrow annulus around the unit circle.
- The oscillation of the modulus on the circle is characterized.

## Abstract

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 $$\left \{z \in {\Bbb C}: 1 - \frac{c_1}{n} < |z| < 1 + \frac{c_1}{n} \right \}\,.$$