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Notation. The following symbols are used for the entities mentioned against them: p, pr\ any prime. S(x, p) : the set of integers less than x each divisible by p and free of prime divisors greater than p. T(x, p) : the set of integers less than x each free of prime divisors greater than p. N\K\: the number of members of K, where K denotes any finite set of integers. ^ ( 0 : X)p^< iX/P) where p runs through primes.

On the number of positive integers less than 𝑥 and free of prime divisors greater than 𝑥^{𝑐}