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.
(An official version is available online.)
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.