Title: On truncated pretentious distances of multiplicative functions, and applications
Abstract: We will be interested in recent progress in the study of correlations of multiplicative functions and on Eliott’s conjecture. In particular, I will present an upper bound, due to Klurman, Mangerel, and Teräväinen, for correlations of multiplicative functions in terms of their truncated pretentious distance. I will then describe a construction, introduced by the same authors, of multiplicative functions given a Furstenberg system. Motivated by these ideas, I will also present new results concerning classes of multiplicative functions satisfying Elliott’s conjecture, obtained as consequences of the previously mentioned upper bound. These results illustrate how quantitative control via pretentious distances on short scales can be used to understand statistical properties of specific multiplicative functions.