zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The distribution of totients. (English) Zbl 0914.11053

This admirable paper improves our knowledge on the set of values of Euler’s φ-function, called totients, considerably. Let V(x) denote the number of nx which are values of φ, let A(n) be the number of solutions m of the equation φ(m)=n, and let V k (x) be the number of nx for which A(n)=k. The main results of the paper are as follows.

Theorem 1:

V(x)=x logxexp(C(log 3 x-log 4 x) 2 +Dlog 3 x-(D+1 2-2C)log 4 x+O(1))

(C and D 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 V(x) for the first time.

Theorem 2. If there is a number d with A(d)=k, then

V k (x) ε d -1+ε V(x)(xx 0 (k))·

The author provides a list of the numbers m k , the smallest m such that A(m)=k, for 2k1000. In 1907, Carmichael conjectured that for no m the equation A(m)=1 holds. The authors show that A(m)=1 implies m10 10 10 and, as a corollary to theorems 1 and 2, that Carmichael’s conjecture is equivalent to the statement lim inf x V 1 (x)/V(x)=0.

It was conjectured by Sierpiński, that for all k2 there are numbers m with A(m)=k. A. Schinzel [Acta Arith. 7, 1-8 (1961; Zbl 0101.27902)] proved this, assuming the well-known hypothesis H, stated by Schinzel and Sierpiński. The present author derives Sierpiński’s conjecture from the prime k-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 log 2 n·1 1-ρ, where ρ=0·54 is the unique number such that F(ρ)=1, where F(x)= n=1 a n x n , a n =(n+1)log(n+1)-nlogn-1.

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.

11N64Characterization of arithmetic functions
11A25Arithmetic functions, etc.
11N37Asymptotic results on arithmetic functions