Publications of (and about) Paul Erdös
Autor: Erdös, Paul
Title: On the normal number of prime factors of p-1 and some related problems concerning Euler's \phi-function. (In English)
Source: Q. J. Math., Oxf. Ser. 6, 205-213 (1935).
Review: The main results of this paper are as follows.
I. The normal number of (different) prime factors of p-1 (where p is a prime) is log log n, i.e. if \epsilon > 0 is given, then for all but o(n/ log n) primes p \leq n, the number of prime factors of p-1 lies between (1-\epsilon) log log n and (1+\epsilon) log log n.
II. The number of integers m \leq n which are representable as \phi (m') (where \phi is Euler's function) is O(n(log n)\epsilon -1), for any \epsilon > 0
III. There exist infinitely many integers m which are representable as \phi (m') in more than mC ways, where C is an absolute constant.
For the proof of I, an upper bound for the number of primes p \leq n for which (p-1)/a is a prime is obtained by Brun's method, and from it is deduced an upper bound for the number of primes p \leq n for which p-1 has exactly k prime factors. For II, the author succeeds in dividing the integers m' with \phi (m') \leq n into two classes, in such a way that the first class contains only O(n/(log n)\epsilon -1) numbers m', and that for the second class \phi (m') has at least 20 log log n prime factors and so can be shown to assume only o(n/ log n) different values. The proof of III cannot be summarised here.
Classif.: * 11N25 Distribution of integers with specified multiplicative constraints
11A25 Arithmetic functions, etc.
Index Words: Algebra, number theory
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