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arXiv:1103.4451 [math.NT]AbstractReferencesReviewsResources

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

Giovanni Coppola

Published 2011-03-23, updated 2011-05-23Version 2

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 $n$).

Comments: The paper has been withdrawn by the Author because of a crucial error in Lemma 2, due to the wrong Lemma A
Categories: math.NT
Subjects: 11N37, 11N25
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