Abstract
Let be -bounded multiplicative functions, and let be shifts. We consider correlation sequences of the form where are numbers going to infinity as and is a generalized limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely, that these sequences are the uniform limit of periodic sequences . Furthermore, if the multiplicative function “weakly pretends” to be a Dirichlet character , the periodic functions can be chosen to be -isotypic in the sense that whenever is coprime to the periods of and , while if does not weakly pretend to be any Dirichlet character, then 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.
Citation
Terence Tao. Joni Teräväinen. "The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures." Duke Math. J. 168 (11) 1977 - 2027, 15 August 2019. https://doi.org/10.1215/00127094-2019-0002
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