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- W2049449509 abstract "Reference [3] contains a discussion of the problem and of some earlier proofs. Our proof is more conceptual than those listed in the bibliography and is based on one simple observation: a multiplicative f: N -1R+ can be extended uniquely to a multiplicative function on Q+ . If f is increasing on N, then it is also increasing on Q I and can be extended to an increasing on R I, which is almost totally multiplicative. It is then easy to prove that f is continuous and totally multiplicative, and it is well known that a continuous endomorphism of Ri + is necessarily raising to a power. For the proof, we need the following well-known fact from real analysis: An increasing from the reals to themselves has a countable set of discontinuities. Except for this, the paper is self-contained. First, whenever we write a rational number p/q E Q I, it will be assumed that p and q are positive integers such that (p, q) = 1. Two rational numbers a = p/q and ,B = r/s are relatively prime, written (a, /3) = 1, if p, q, r, and s are relatively prime in pairs. A f: Q R D + is called multiplicative if f(ao/) = f(a)f(,B) whenever (a, 8) = 1." @default.
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- W2049449509 date "1989-06-01" @default.
- W2049449509 modified "2023-09-26" @default.
- W2049449509 title "Monotone Multiplicative Functions" @default.
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- W2049449509 doi "https://doi.org/10.2307/2323973" @default.
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