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The distribution of totients

Author: Kevin Ford
Journal: Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34
MSC (1991): Primary 11A25, 11N64; Secondary 11N35
Published electronically: April 27, 1998
MathSciNet review: 1617448
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Abstract: 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 Sierpinski. 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.

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Additional Information

Kevin Ford
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, TX 78712

Keywords: Euler's function, totients, distributions, Carmichael's conjecture, Sierpi\'{n}ski's conjecture
Received by editor(s): August 13, 1997
Published electronically: April 27, 1998
Communicated by: Hugh Montgomery
Article copyright: © Copyright 1998 American Mathematical Society

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