Abstract
We study the asymptotic behaviour of higher order correlations
as a function of the parameters and , where are bounded multiplicative functions, are integer shifts, and is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all if does not (weakly) pretend to be a twisted Dirichlet character , and behave asymptotically like a multiple of otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the parameter is averaged out and one can set . Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the -point Chowla conjecture for odd or equal to for all scales outside of a set of zero logarithmic density.
Citation
Terence Tao. Joni Teräväinen. "The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures." Algebra Number Theory 13 (9) 2103 - 2150, 2019. https://doi.org/10.2140/ant.2019.13.2103
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