On the symmetry of the Liouville function in almost all short intervals

Abstract

We prove a kind of "almost all symmetry" result for the Liouville function $\lambda (n):=(-1{)}^{\Omega (n)}$, giving non-trivial bounds for its "symmetry integral", say $I_{\lambda }(N,h)$ : we get $I_{\lambda }(N,h)\ll Nh{L}^3+N{h}^{21/20}$, with $L:=\log N$. We also give similar results for other related arithmetic functions, like the Möbius function $\mu (n)$ ($=\lambda (n)$ on square-free $n$).

The paper has been withdrawn by the Author because of a crucial error in Lemma 2, due to the wrong Lemma A

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