Abstract
Let ${\it\lambda}$ and ${\it\mu}$ denote the Liouville and Möbius functions, respectively. Hildebrand showed that all eight possible sign patterns for $({\it\lambda}(n),{\it\lambda}(n+1),{\it\lambda}(n+2))$ occur infinitely often. By using the recent result of the first two authors on mean values of multiplicative functions in short intervals, we strengthen Hildebrand’s result by proving that each of these eight sign patterns occur with positive lower natural density. We also obtain an analogous result for the nine possible sign patterns for $({\it\mu}(n),{\it\mu}(n+1))$. A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.
Highlights
We strengthen some results on the sign patterns of the Liouville function λ, as well as obtain new results on the Mobius function μ
Set k to be the odd number k := 100 log log X log log log X. The reason for this somewhat strange choice is thatk will be significantly larger than log10 X, while k remains significantly smaller than log log X
We write γ (k) := p1 + · · · + pk for the endpoint of such a path, which is automatically odd since k and the p1, . . . , pk are odd, and by abuse of notation write γ ⊂ GX if the path γ lies in GX, or equivalently that pi |n + p1 + · · · + pi−1 for all i = 1, . . . , k
Summary
We strengthen some results on the sign patterns of the Liouville function λ, as well as obtain new results on the Mobius function μ. It is plausible that using arguments similar to the ones here, one can show that these sixteen sign patterns occur for n in a set of positive lower density. In the case of the Liouville function, one could show (as was done in [12]) that if the analogous claim (λ(n), λ(n + 1)) = (+1, −1) occurs with zero density, there would often exist long chains n, n + 1, .
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