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 number of solutions of φ(x)=m. (English) Zbl 0978.11053

Let A(m) denote the number of integers n for which φ(n)=m, let V k (x) denote the number of mx for which A(m)=k and let V(x)= k1 V k (x). Carmichael conjectured that A(m)1 for all m (that is, V 1 (x)=0); Sierpiński conjectured that for every integer k2, there does exist m with A(m)=k.

In this delightful paper, Ford proves Sierpiński’s long elusive conjecture, even proving the surprising result that V k (x) k V(x) for x>x k , whenever k2. He had already proved extraordinarily strong estimates for V(x) in [Ramanujan J. 2, 67-151 (1998; Zbl 0914.11053)]. The author and S. Konyagin [Number Theory in Progress, Vol. 2, 795-803 (1999; Zbl 0931.11032)] had already proved Sierpiński’s conjecture for even k, but the odd k case seems to be considerably more difficult.

The starting point for Ford’s method is an easy proof of Sierpiński’s conjecture under the assumption of the prime triplets conjecture; for if A(m)=k and p is a large prime such that dp+1 is prime for d dividing 2m only when d=2 or 2m, then A(2mp)=k+2. Of course we do not know whether the prime triplets conjecture is true, but we do know, from sieve methods, that we can determine k-tuples of “almost primes”, that is integers with few and only large prime factors. Thus Ford suitably modifies the above construction to incorporate such k-tuples. This requires considerable ingenuity and technical progress, making this proof an impressive achievement.

11N64Characterization of arithmetic functions
11N37Asymptotic results on arithmetic functions
11A25Arithmetic functions, etc.
11N60Distribution functions (additive and positive multiplicative functions)