Phénomène de Moser-Newman pour les nombres sans facteur carré

Abstract

Let μ denotes the Mobius function and t = (t(n))n∈N be the Thue-Morse sequence, defined by t(n) = +1 if the number of 1 in the dyadic representation of n is even and t(n) = −1 otherwise. The aim of this work is to give an asymptotic formula for S(N) = Σn<N μ2(n)t(n) and to prove that S(N) < 0 for N big enough. This shows that square-free integers provide a first example of non-linear Moser-Newman phenomenon. Our method gives a similar result for z-th power free integers.