Möbius-Walsh correlation bounds and an estimate of Mauduit and Rivat

Abstract

We establish small correlation bounds for the Möbius function and the Walsh system, answering affirmatively a question posed by G. Kalai [Ka] . The argument is based on generalizing the approach of Mauduit and Rivat [M-R] in order to treat Walsh functions of "large weight", while the "small weight" case follows from recent work due to B. Green [Gr] . The conclusion is an estimate uniform over the full Walsh system. A similar result also holds for the Liouville function.