Dynamical models for Liouville and obstructions to further progress on sign patterns

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Abstract

We define a class of dynamical systems by modifying a construction due to Tao, which includes certain Furstenburg limits arising from the Liouville function. Most recent progress on the Chowla conjectures and sign patterns of the Möbius and Liouville functions uses methods that apply to any dynamical system in this class. Hence dynamical systems in this class with anomalous local behavior present obstructions to further progress on these problems by the same techniques. We construct straightforward examples of dynamical systems in this class based on polynomial phases and calculate the resulting obstruction. This requires explicit bounds for the number of sign patterns arising in a certain way from polynomials, which is elementary but not completely trivial.

Original languageEnglish (US)
Pages (from-to)1-15
Number of pages15
JournalJournal of Number Theory
Volume213
DOIs
StatePublished - Aug 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Keywords

  • Dynamical systems
  • Liouville function
  • Sign patterns

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