SIGN PATTERNS of the LIOUVILLE and MÖBIUS FUNCTIONS

Kaisa Matomäki, Maksym RadziwiŁŁ, Terence Tao

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

Abstract

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.

Original languageEnglish (US)
Article numbere14
JournalForum of Mathematics, Sigma
Volume4
DOIs
StatePublished - 2016

ASJC Scopus subject areas

  • Analysis
  • Theoretical Computer Science
  • Algebra and Number Theory
  • Statistics and Probability
  • Mathematical Physics
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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