Abstract
Letting $\Omega(n)$ denote the number of prime factors of $n$ counted with multiplicity, Rivat, Sárközy and Stewart (1999) proved a result regarding maximal cardinalities of sets ${\cal A},{\cal B}\subset\{1,\ldots,N\}$ so that for every $a\in{\cal A}$ and $b\in{\cal B}$, $\Omega(a+b)$ is even. This paper extends their work in several directions. The role of $\lambda(n)=(-1)^{\Omega(n)}$ is generalized to all non-constant completely multiplicative functions $f:\mathbb{N}\rightarrow \{-1,1\}$. Rather than just $\Omega$ being even on ${\cal A}+{\cal B}$, we extend the result to all possible parities of $\Omega$ on ${\cal A}$, ${\cal B}$, and ${\cal A}+{\cal B}$. Furthermore, we prove that many such pairs $({\cal A},{\cal B})$ exist. Results from Ramsey theory and extremal graph theory are used.
Citation
Christian Elsholtz. David S. Gunderson. "Congruence properties of multiplicative functions on sumsets and monochromatic solutions of linear equations." Funct. Approx. Comment. Math. 52 (2) 263 - 281, June 2015. https://doi.org/10.7169/facm/2015.52.2.6
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