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On multiplicative functions which are additive on almost primes
University of Debrecen/ Debreceni Egyetem - Public 2018 Publicationes mathematicae Vol.92 No.3
<P>In 1992, C. Spiro showed that if a multiplicative function f satisfies f (p + q) = f (p) + f (q) for all primes p and q, and f (p(0)) does not vanish at some prime p(0), then f is the identity function. In this article, we extend Spiro's result to products of exactly k prime factors with multiplicity, which are called k-almost primes. That is, if a multiplicative function f satisfies f (P + Q) = f (P) + f (Q) for all k-almost primes P and Q, and f (n(0)) does not vanish at some k-almost prime n(0), then f is the identity function.</P>