This admirable paper improves our knowledge on the set of values of Euler’s -function, called totients, considerably. Let denote the number of which are values of , let be the number of solutions of the equation , and let be the number of for which . The main results of the paper are as follows.
( and are well defined positive constants).
This improves the result of H. Maier and C. Pomerance [Acta Arith. 49, 263-275 (1988; Zbl 0638.10045)] and determines the true order of for the first time.
Theorem 2. If there is a number with , then
The author provides a list of the numbers , the smallest such that , for . In 1907, Carmichael conjectured that for no the equation holds. The authors show that implies and, as a corollary to theorems 1 and 2, that Carmichael’s conjecture is equivalent to the statement .
It was conjectured by Sierpiński, that for all there are numbers with . A. Schinzel [Acta Arith. 7, 1-8 (1961; Zbl 0101.27902)] proved this, assuming the well-known hypothesis , stated by Schinzel and Sierpiński. The present author derives Sierpiński’s conjecture from the prime -tuples conjecture. He announces an unconditional proof, based on results on almost primes.
In theorem 12, the author determines the normal number of prime factors (with and without multiplicities) of totients. Roughly speaking, this is equal to , where is the unique number such that , where , .
It is almost impossible to describe briefly all results of this paper, or to give an impression of the methods of proof (sieve arguments, geometric ideas, analytic tools). One has to look forward to seeing further papers of the author.