Abstract
Let \({f : \mathbb{N} \to \mathbb{C}}\) be a multiplicative function satisfying f(p 0) ≠ 0 for at least one prime number p 0, and let k ≥ 2 be an integer. We show that if the function f satisfies f(p 1 + p 2 + . . . + p k ) = f(p 1) + f(p 2) + . . . + f(p k ) for any prime numbers p 1, p 2, . . . ,p k then f must be the identity f(n) = n for each \({n \in \mathbb{N}}\). This result for k = 2 was established earlier by Spiro, whereas the case k = 3 was recently proved by Fang. In the proof of this result for k ≥ 6 we use a recent result of Tao asserting that every odd number greater than 1 is the sum of at most five primes.
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Dubickas, A., Šarka, P. On multiplicative functions which are additive on sums of primes. Aequat. Math. 86, 81–89 (2013). https://doi.org/10.1007/s00010-012-0156-8
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DOI: https://doi.org/10.1007/s00010-012-0156-8