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

DOI:
https://doi.org/10.1090/S1079-6762-98-00043-2

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 Sierpiński. 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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- R. D. Carmichael,
*On Euler’s $\phi$-function*, Bull. Amer. Math. Soc. **13** (1907), 241–243.
- ---,
*Note on Euler’s $\phi$-function*, Bull. Amer. Math. Soc. **28** (1922), 109–110.
- L. E. Dickson,
*A new extension of Dirichlet’s theorem on prime numbers*, Messenger of Math. **33** (1904), 155–161.
- P. Erdös,
*On the normal number of prime factors of $p-1$ and some related problems concerning Euler’s $\phi$-function*, Quart. J. Math. Oxford (1935), 205–213.
- ---,
*Some remarks on Euler’s $\phi$-function and some related problems*, Bull. Amer. Math. Soc. **51** (1945), 540–544.
- ---,
*Some remarks on Euler’s $\phi$-function*, Acta Arith. vol 4 (1958), 10–19.
- P. Erdös and R. R. Hall,
*On the values of Euler’s $\phi$-function*, Acta Arith. **22** (1973), 201-206.
- ---,
*Distinct values of Euler’s $\phi$-function*, Mathematika **23** (1976), 1–3.
- P. Erdös and C. Pomerance,
*On the normal number of prime factors of $\phi (n)$*, Rocky Mountain J. of Math. **15** (1985), 343–352.
- K. Ford,
*The distribution of totients*, The Ramanujan J. **2** (1998), no. 1-2 (to appear).
- ---,
*The number of solutions of $\phi (x)=m$*, Annals of Math. (to appear).
- K. Ford and S. Konyagin,
*On a conjecture of Sierpiński concerning the sum of divisors function*, Proceedings of the International Number Theory Conference dedicated to Professor Andrzej Schinzel, Zakopane, Poland (to appear).
- R. R. Hall and G. Tenenbaum,
*Divisors*, Cambridge University Press, 1988.
- G. H. Hardy and S. Ramanujan,
*The normal number of prime factors of a number $n$*, Quart. J. Math. **48** (1917), 76–92.
- V. Klee,
*On a conjecture of Carmichael*, Bull. Amer. Math. Soc. **53** (1947), 1183–1186.
- H. Maier and C. Pomerance,
*On the number of distinct values of Euler’s $\phi$-function*, Acta Arith. **49** (1988), 263–275.
- P. Masai and A. Valette,
*A lower bound for a counterexample to Carmichael’s Conjecture*, Boll. Un. Mat. Ital. A (6) **1** (1982), 313–316.
- S. Pillai,
*On some functions connected with $\phi (n)$*, Bull. Amer. Math. Soc. **35** (1929), 832–836.
- C. Pomerance,
*On the distribution of the values of Euler’s function*, Acta Arith. **47** (1986), 63–70.
- ---,
*Problem 6671*, Amer. Math. Monthly **98** (1991), 862.
- A. Schinzel,
*Sur l’équation $\phi (x)=m$*, Elem. Math. **11** (1956), 75–78.
- ---,
*Remarks on the paper “Sur certaines hypothèses concernant les nombres premiers”*, Acta Arith. **7** (1961/62), 1–8.
- A. Schinzel and W. Sierpiński,
*Sur certaines hypothèses concernant les nombres premiers*, Acta Arith. **4** (1958), 185–208.
- A. Schlafly and S. Wagon,
*Carmichael’s conjecture on the Euler function is valid below $10^{10,000,000}$*, Math. Comp. **63** (1994), 415–419.

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Additional Information

**Kevin Ford**

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

MR Author ID:
325647

ORCID:
0000-0001-9650-725X

Email:
ford@math.utexas.edu

Keywords:
Euler’s function,
totients,
distributions,
Carmichael’s conjecture,
Sierpiński’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