# Littlewood polynomial

In mathematics, a **Littlewood polynomial** is a polynomial all of whose coefficients are +1 or −1.
**Littlewood's problem** asks how large the values of such a polynomial must be on the unit circle in the complex plane. The answer to this would yield information about the autocorrelation of binary sequences.
They are named for J. E. Littlewood who studied them in the 1950s.

## Definition

A polynomial

is a *Littlewood polynomial* if all the . Let ||*p*|| denote the supremum of |*p*(*z*)| on the unit circle. *Littlewood's problem* asks for constants *c*_{1} and *c*_{2} such that there are infinitely many *p*_{n} , of increasing degree *n*, such that

The Rudin-Shapiro polynomials provide a sequence satisfying the upper bound with . No sequence is known (as of 2008) which satisfies the lower bound.

## See also

## References

- Peter Borwein (2002).
*Computational Excursions in Analysis and Number Theory*. Springer-Verlag, 2-5,121-132. ISBN 0-387-95444-9. - J.E. Littlewood (1968).
*Some problems in real and complex analysis*. D.C. Heath.