Congruence properties of multiplicative functions on sumsets and monochromatic solutions of linear equations

Abstract

Letting Ω(n) denote the number of prime factors of n counted with multiplicity, Rivat, Sárközy and Stewart (1999) proved a result regarding maximal cardinalities of sets A, B ⊂ {1, . . . , N } so that for every a ∈ A and b ∈ B, Ω(a + b) is even.This paper extends their work in several directions. The role of λ(n) = (−1) Ω(n) is generalized to all non-constant completely multiplicative functions f : N → {−1, 1}. Rather than just Ω being even on A + B, we extend the result to all possible parities of Ω on A, B, and A + B. Furthermore, we prove that many such pairs (A, B) exist. Results from Ramsey theory and extremal graph theory are used.

Multiplicative functions in short intervals

“…Using Halász's theorem one can show that if f (p)<0 1/p = ∞ and f (n) = 0 for a positive proportion of the integers n then the non-zero values of f (n) are half of the time positive and half of the time negative (see [30,Lemma 2.4] or [7,Lemma 3.3]). Since we expect f (n) and f (n + 1) to behave independently this suggests that, for non-vanishing f such that f (p)<0 1/p = ∞, there should be about x/2 sign changes among integers n ≤ x.…”