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

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

2011-03-23

arxiv.org/abs/1103.4451v2

Mathematics

Number Theory

The paper has been withdrawn by the Author because of a crucial error in Lemma 2, due to the wrong Lemma A

Scientific paper

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 $I_{\lambda}(N,h)\ll NhL^3+Nh^{21/20}$, with $L:=\log N$. We also give similar results for other related arithmetic functions, like the M\"{o}bius function $\mu(n)$ ($=\lambda(n)$ on square-free $n$).

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