Luca 2011 arithmetic function arising carmichael s conjecture

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lemma_2_1: For x sufficiently large and every squarefree d <= x, the number S(x; d) of n with phi(n) <= x a multiple of d is at most B_{omega(d)} (C_1 log log x)^{omega(d)} x (log log x)^2 / d, with B_k the Bell numbers.

theorem_1_1: For each fixed epsilon > 0 and almost all n, the number F(n) of m with phi(m) = phi(n) lies strictly between K(n)^{1/2 - epsilon} and K(n)^{3/2 + epsilon}, where K(x) = (log x)^{(log log x)(log log log x)}.

theorem_1_2: For x >= 20, the sum over n <= x of the square of Omega(phi(n) + 1) minus omega(phi(n) + 1) is O(x (log log log x)^5 / log log x), so phi(n) + 1 is squarefree for almost all n.

theorem_1_3: For fixed delta in (0, 1), a v <= x with fewer than (log x)^{1 - delta} distinct prime factors up to (log x)^{1 + delta} has at most x/L(x)^{1 + delta + o(1)} preimages under phi, and for any v <= x at most that many preimages m have omega(m) <= log x/(log log x)^{2 + delta}.

theorem_2_1: For squarefree d <= x with d >= exp((log x)^{1/log_3 x}), S(x; d) is at most x/d^{eta + o(1)} when the roundness of d is at most 1 - eta, and at most x/L(d)^{1 + o(1)} uniformly in d.

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Luca, Florian and Pollack, Paul, An arithmetic function arising from Carmichael's conjecture. J. Théor. Nombres Bordeaux 23 (2011), no. 3, 697--714, DOI 10.5802/jtnb.783. The copy read for this card is the journal's PDF from its cedram archive, whose cover page prints "© Société Arithmétique de Bordeaux, 2011, tous droits réservés.", every other right reserved.

Source: https://jtnb.centre-mersenne.org/item/10.5802/jtnb.783/.

Let $F(n)$ be the number of $m$ with $\phi(m)=\phi(n)$; Carmichael's conjecture is that $F(n)\ge2$ always. The paper studies the normal size of $F$. Its main result, Theorem 1.1 (p. 698), is that for each fixed $\epsilon>0$ and all $n$ outside a set of density zero, $K(n)^{1/2-\epsilon}<F(n)<K(n)^{3/2+\epsilon}$, where $K(x)=(\log x)^{(\log\log x)(\log\log\log x)}$. The engine is Lemma 2.1 (p. 701), a uniform upper bound for the number $S(x;d)$ of $n$ with $\phi(n)\le x$ divisible by a squarefree $d$, combined with the Erdős--Pomerance normal order of $\omega(\phi(n))$. Theorem 2.1 (p. 702) records what the lemma gives for squarefree $d\ge\exp((\log x)^{1/\log_3x})$, $\log_3$ the triple logarithm, where $\omega(d)$ may be large. As an application, Theorem 1.2 (p. 699) shows that the second moment of $\Omega(\phi(n)+1)-\omega(\phi(n)+1)$ over $n\le x$ is $\ll x(\log\log\log x)^5/\log\log x$ for $x\ge20$, so $\phi(n)+1$ is squarefree for almost all $n$.

The introduction (p. 698) recalls the results on large values of $F$: Erdős's 1935 theorem that $F(n)>n^c$ infinitely often for some $c>0$, the value $c=0.7038$ that the paper attributes to Baker and Harman's work, the conjecture that every $c<1$ is permissible, and Pomerance's bound (1.1), $\max_{n\le x}F(n)\le x/L(x)^{1+o(1)}$ with $L(x)=x^{\log\log\log x/\log\log x}$, with equality under a hypothesis on smooth shifted primes. None of these is proved in the paper. Theorem 1.3 (p. 700), for fixed $0<\delta<1$, gives a necessary condition for a value $v\le x$ to have more than $x/L(x)^{1+\delta+o(1)}$ preimages: at least $(\log x)^{1-\delta}$ distinct prime factors up to $(\log x)^{1+\delta}$. It also shows that at most $x/L(x)^{1+\delta+o(1)}$ preimages of any $v\le x$ have at most $\log x/(\log\log x)^{2+\delta}$ distinct prime factors.

Read status: claims checked for the results linked below, statements read clause by clause on the page images of the print; the proofs of Lemma 2.1, Theorem 2.1 and Theorem 1.3 followed, those of Theorems 1.1 and 1.2 read for structure. Nothing here is independently reviewed.

Results.

• Theorem 1.1 (p. 698): for almost all $n$, $F(n)$ lies between $K(n)^{1/2-\epsilon}$ and $K(n)^{3/2+\epsilon}$.
• Theorem 1.2 (p. 699): the second-moment bound (1.2), so $\phi(n)+1$ is squarefree for almost all $n$.
• Theorem 1.3 (p. 700): for fixed $0<\delta<1$, a value $v\le x$ with fewer than $(\log x)^{1-\delta}$ distinct prime factors up to $(\log x)^{1+\delta}$ has at most $x/L(x)^{1+\delta+o(1)}$ preimages, and any $v\le x$ has at most that many preimages $m$ with $\omega(m)\le\log x/(\log\log x)^{2+\delta}$.
• Lemma 2.1 (p. 701): the uniform bound for $S(x;d)$.
• Theorem 2.1 (p. 702): bounds for $S(x;d)$ when $d\le x$ is squarefree and $d\ge\exp((\log x)^{1/\log_3x})$.

Bears on. #821: the paper proves no lower bound for the number of preimages of a value. Its introduction (p. 698) cites Erdős's theorem that $F(n)>n^c$ infinitely often for some $c>0$ and the value $c=0.7038$ that it attributes to Baker and Harman's work, and records the conjecture that any $c<1$ is permissible, phrased for $F(n)=\#\phi^{-1}(\phi(n))$ rather than for the number of preimages of $n$. Theorem 1.3 (p. 700) gives a necessary condition on a value $v\le x$ with more than $x/L(x)^{1+\delta+o(1)}$ preimages. It decides nothing about the problem.

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