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The distribution of totients. (English) Zbl 0914.11053

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

Theorem 1:

$V\left(x\right)=\frac{x}{logx}exp\left(C{\left({log}_{3}x-{log}_{4}x\right)}^{2}+D{log}_{3}x-\left(D+\frac{1}{2}-2C\right){log}_{4}x+O\left(1\right)\right)$

($C$ 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 $V\left(x\right)$ for the first time.

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

${V}_{k}\left(x\right){\gg }_{\epsilon }{d}^{-1+\epsilon }V\left(x\right)\phantom{\rule{2.em}{0ex}}\left(x\ge {x}_{0}\left(k\right)\right)·$

The author provides a list of the numbers ${m}_{k}$, the smallest $m$ such that $A\left(m\right)=k$, for $2\le k\le 1000$. In 1907, Carmichael conjectured that for no $m$ the equation $A\left(m\right)=1$ holds. The authors show that $A\left(m\right)=1$ implies $m\ge {10}^{{10}^{10}}$ and, as a corollary to theorems 1 and 2, that Carmichael’s conjecture is equivalent to the statement ${lim inf}_{x\to \infty }{V}_{1}\left(x\right)/V\left(x\right)=0$.

It was conjectured by Sierpiński, that for all $k\ge 2$ there are numbers $m$ with $A\left(m\right)=k$. 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 ${log}_{2}n·\frac{1}{1-\rho }$, where $\rho =0·54\cdots$ is the unique number such that $F\left(\rho \right)=1$, where $F\left(x\right)={\sum }_{n=1}^{\infty }{a}_{n}{x}^{n}$, ${a}_{n}=\left(n+1\right)log\left(n+1\right)-nlogn-1$.

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.

MSC:
 11N64 Characterization of arithmetic functions 11A25 Arithmetic functions, etc. 11N37 Asymptotic results on arithmetic functions