Everett Howe: A new proof of Erdős's theorem on monotone multiplicative functions, Amer. Math. Monthly 93 (1986) 593–595, MR 87k:11010, Zbl 0614.10001.

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I provide a new proof of the fact that every increasing multiplicative function from the positive integers to the reals is of the form f(n)=na for some real a>1.