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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 012.14905

**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 m^{C} 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.

**Reviewer: ** Davenport

**Classif.: ** * 11N25 Distribution of integers with specified multiplicative constraints

11A25 Arithmetic functions, etc.

**Index Words: ** Algebra, number theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag