### On Correlations of Liouville-like functions

Linfoot Number Theory Seminar

22nd May 2024, 11:00 am – 12:00 pm

Fry Building, 2.04

Let A be a set of mutually coprime positive integers, satisfying \sum_{a \in A}\frac{1}{a} = \infty. Define the (possibly non-multiplicative) "Liouville-like" functions \lambda_{A}(n) = (-1)^{|\{a : a | n, a \in A \}|} or (-1)^{|\{a : a^\nu \parallel n, a \in A, \nu \in \mathbb{N}\}|}. We show that \lim{x\to\infty}\frac{1}{x}\sum_{n \leq x} \lambda_A(n) = 0 holds, answering a question of de la Rue. We also show that if $A \cap \mathbb{P} has relative density 0 in \mathbb{P}, the k-point correlations of \lambda_{A} satisfies \lim_{x \to \infty}\frac{1}{x}\sum_{n\leq x}\lambda_A(a_1n+h_1)\cdots\lambda_A(a_kn+h_k) = 0, where k \geq 2, a_ih_j \neq a_jh_i for all 1 \leq i < j \leq k, extending a recent result of O. Klurman, A. P. Mangerel, and J. Teräväinen.

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