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This paper is an announcement of many new results concerning the set of totients, i.e. the set of values taken by Euler's $\phi $-function. The main functions studied are $V(x)$, the number of totients not exceeding $x$, $A(m)$, the number of solutions of $\phi (x)=m$ (the ``multiplicity'' of $m$), and $V_{k}(x)$, the number of $m\le x$ with $A(m)=k$. The first of the main results of the paper is a determination of the true order of $V(x)$. It is also shown that for each $k\ge 1$, if there is a totient with multiplicity $k$, then $V_{k}(x) \gg V(x)$. We further show that every multiplicity $k\ge 2$ is possible, settling an old conjecture of Sierpi\'{n}ski. An older conjecture of Carmichael states that no totient has multiplicity 1. This remains an open problem, but some progress can be reported. In particular, the results stated above imply that if there is one counterexample, then a positive proportion of all totients are counterexamples. Determining the order of $V(x)$ and $V_{k}(x)$ also provides a description of the ``normal'' multiplicative structure of totients. This takes the form of bounds on the sizes of the prime factors of a pre-image of a typical totient. One corollary is that the normal number of prime factors of a totient $\le x$ is $c\log \log x$, where $c\approx 2.186$. Lastly, similar results are proved for the set of values taken by a general multiplicative arithmetic function, such as the sum of divisors function, whose behavior is similar to that of Euler's function.

Copyright 1998 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 04 (1998), pp. 27-34
• Publisher Identifier: S 1079-6762(98)00043-2
• 1991 Mathematics Subject Classification. Primary 11A25, 11N64; Secondary 11N35
• Key words and phrases. Euler's function, totients, distributions, Carmichael's conjecture, Sierpinski's conjecture
• Received by the editors August 13, 1997
• Posted on April 27, 1998
• Communicated by Hugh Montgomery
• Comments (When Available)

Kevin Ford

Department of Mathematics, University of Texas at Austin, Austin, TX 78712

E-mail address: ford@math.utexas.edu