Title: Multiplicative functions and sumsets
David S. Gunderson (University of Manitoba)
joint work with Christian Elsholtz (Royal Holloway, University of London)
For a positive integer n, the number of prime factors of n (not necessarily
distinct) is denoted by Ω. In 1999, Rivat, Sárközy and Stewart 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.
Using applications from arithmetic Ramsey theory and extremal graph theory, their work can be extended in several directions. The role of Liousville
λ-function, given by λ(n) = (−1)Ω(n) , is generalized to all non-constant completely multiplicative functions f from the natural numbers to {−1, 1}. The
results are also extended to all possible parities of Ω on A, B, and A + B, and
in each case, the pairs (A, B) occur with positive density.
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