A1 Refereed original research article in a scientific journal
THE STRUCTURE OF LOGARITHMICALLY AVERAGED CORRELATIONS OF MULTIPLICATIVE FUNCTIONS, WITH APPLICATIONS TO THE CHOWLA AND ELLIOTT CONJECTURES
Authors: Terence Tao, Joni Teräväinen
Publisher: DUKE UNIV PRESS
Publication year: 2019
Journal:Duke Mathematical Journal
Journal name in sourceDUKE MATHEMATICAL JOURNAL
Journal acronym: DUKE MATH J
Volume: 168
Issue: 11
First page : 1977
Last page: 2027
Number of pages: 51
ISSN: 0012-7094
DOI: https://doi.org/10.1215/00127094-2019-0002
Abstract
Let g(0), ..., g(k) : N -> D be 1-bounded multiplicative functions, and let h(0), ..., h(k) is an element of Z be shifts. We consider correlation sequences f : N -> Z of the formf(a) :=lim(m ->infinity) 1/log omega(m) Sigma x(m)/omega(m)<= n <= x(m) g(0)(n +ah(0))...g(k)(n + ah(k))/n,where 1 <= omega(m) <= x(m) are numbers going to infinity as m -> infinity and (lim) over bar is a generalized limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely, that these sequences f are the uniform limit of periodic sequences fi. Furthermore, if the multiplicative function g(0) ... g(k) "weakly pretends" to be a Dirichlet character chi, the periodic functions fi can be chosen to be chi- isotypic in the sense that f(i) (ab) = f(i)(a) chi(b) whenever b is coprime to the periods of f(i) and chi, while if g(0) ... g(k) does not weakly pretend to be any Dirichlet character, then f must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three and of the Möbius function of length up to four.
Let g(0), ..., g(k) : N -> D be 1-bounded multiplicative functions, and let h(0), ..., h(k) is an element of Z be shifts. We consider correlation sequences f : N -> Z of the formf(a) :=lim(m ->infinity) 1/log omega(m) Sigma x(m)/omega(m)<= n <= x(m) g(0)(n +ah(0))...g(k)(n + ah(k))/n,where 1 <= omega(m) <= x(m) are numbers going to infinity as m -> infinity and (lim) over bar is a generalized limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely, that these sequences f are the uniform limit of periodic sequences fi. Furthermore, if the multiplicative function g(0) ... g(k) "weakly pretends" to be a Dirichlet character chi, the periodic functions fi can be chosen to be chi- isotypic in the sense that f(i) (ab) = f(i)(a) chi(b) whenever b is coprime to the periods of f(i) and chi, while if g(0) ... g(k) does not weakly pretend to be any Dirichlet character, then f must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three and of the Möbius function of length up to four.
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