The average sensitivity of square-freeness

Abstract

We study combinatorial complexity characteristics of a Boolean function related to a natural number theoretic problem. In particular we obtain an asymptotic formula, having a linear main term, for the average sensitivity of the Boolean function deciding whether a given integer is square-free. This result allows us to derive a quadratic lower bound for the formula size complexity of testing square-free numbers and a linear lower bound on the average decision tree depth. We also obtain lower bounds on the degrees of exact and approximate polynomial representations of this function.