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Abstract: For each natural number N, let p_N denote the nth prime number, and let Omega_N denote the set of positive integers all of whose prime factors are less than or equal to p_N. Let P_N denote the probability measure on Omega_N for which P_N(n) is proportional to n. This measure turns out to have some very useful and interesting properties which are related to the theory of additive arithmetic functions.

After recalling and discussing briefly some seminal results of this theory, such as those of Erdos-Wintner, Kac-Erdos and Hardy-Ramanujan, we will investigate P_N more closely. This will lead us to the Dickman function and "smooth" numbers, which are numbers without large prime factors, and to the Buchstab function and "rough" numbers, which are numbers without small prime factors. These two functions satisfy differential-delay equations. We obtain a new representation of the Buchstab function.