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Published June 21, 2016 | Published + Submitted
Journal Article Open

Sign patterns of the Liouville and Möbius functions


Let λ and µ denote the Liouville and Möbius functions, respectively. Hildebrand showed that all eight possible sign patterns for λ(n), λ(n+1), λ(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 µ(n), µ(n+1). A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.

Additional Information

© The Author(s) 2016. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Received 4 September 2015; accepted 8 April 2016.

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Submitted - 1509.01545.pdf


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