Let $A\left(m\right)$ denote the number of integers $n$ for which $\phi \left(n\right)=m$, let ${V}_{k}\left(x\right)$ denote the number of $m\le x$ for which $A\left(m\right)=k$ and let $V\left(x\right)={\sum}_{k\ge 1}{V}_{k}\left(x\right)$. Carmichael conjectured that $A\left(m\right)\ne 1$ for all $m$ (that is, ${V}_{1}\left(x\right)=0$); Sierpiński conjectured that for every integer $k\ge 2$, there does exist $m$ with $A\left(m\right)=k$.

In this delightful paper, Ford proves Sierpiński’s long elusive conjecture, even proving the surprising result that ${V}_{k}\left(x\right){\gg}_{k}V\left(x\right)$ for $x>{x}_{k}$, whenever $k\ge 2$. He had already proved extraordinarily strong estimates for $V\left(x\right)$ 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 $A\left(m\right)=k$ and $p$ is a large prime such that $dp+1$ is prime for $d$ dividing $2m$ only when $d=2$ or $2m$, then $A\left(2mp\right)=k+2$. 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.

##### MSC:

11N64 | Characterization of arithmetic functions |

11N37 | Asymptotic results on arithmetic functions |

11A25 | Arithmetic functions, etc. |

11N60 | Distribution functions (additive and positive multiplicative functions) |