Sign patterns of the Liouville and Möbius functions

Abstract

<p>Let <img src="https://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20160916133015054-0035:S2050509416000062_inline1.gif" />λ λ

and <img src="https://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20160916133015054-0035:S2050509416000062_inline2.gif" />μ μ

denote the Liouville and Möbius functions, respectively. Hildebrand showed that all eight possible sign patterns for <img src="https://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20160916133015054-0035:S2050509416000062_inline3.gif" />(λ(n),λ(n+1),λ(n+2)) (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 <img src="https://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20160916133015054-0035:S2050509416000062_inline4.gif" />(μ(n),μ(n+1)) (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.<br /></p>