ON THE NORMAL NUMBER OF PRIME FACTORS OF p-l AND SOME RELATED PROBLEMS CONCERNING EULER’S +-FUNCTION

THIS paper is concerned wit,h some problems considered by Hardy and R’amanujan, Titchmarsh, and Pillai. Suppose we are given a set ii of positive integers m. Let N(n,) denote Dhe number of m in t’he interval (0, n). By saying t’hat the norma, number of prime factors of a number m is B(n), we mean Dha’t, as n + co, t,here are only o[N(n)] of the m (< n) for which the number of prime factors does not lie between (1 -&c)B(n) for arbikarily small positive E. We use throughout the following not’ation: N(M, n) denot’es t,he number of integers not exceeding n in the set A!; d(n) is Dhe number of divisors of n.; p = logn, v = loglogn; p, pl, pi,... are prime numbers, and C,, C,:... denote posit’ive const,ants independent of n; m. In t’he first part,, I prove that, if X is t’he set p-1; and so N(n) n/p, then B(n) = Y. I use the method of Brun and also that employed by Hardy and Ramanujan” in their proof that, when N is t’he set of all natural numbers, B(n) = v. I then apply my result t’o a problem of Titchmarsh”f who showed that,, if