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Source. Rafik Zeraoulia, Fixed-Scale Limit Points for the Counting Function of Distinct Euler Totients, preprint, July 2026, in the fifteen-page PDF held by its library source card, Zeraoulia (2026): Theorem 1.1 (physical p. 2), proved through Lemma 2.1 and Theorem 2.2 (§2, p. 3), Lemma 3.1, Proposition 3.2, Theorems 3.3 and 3.4 and Corollary 3.5 (§3, pp. 4–5), with Proposition 6.1 (p. 8) added for its bearing on $c=2$. Physical and numbered pages coincide. Pages 2–5 and 8 were read against the page images; the rest of the preprint (the matched quotient of §4, the block-energy identity and second-moment criterion of §5, the entropy recursion of §6, the log-periodic model of §7 and the computation of §8) was read in text extraction only and is not reconstructed here.

Standing. This is an author-recorded reconstruction of the unconditional part of a self-published, unreviewed preprint. It is not an independent review, changes no status of Problem 416 and assigns no tier. The only external inputs are Ford's Theorem 1 and Chebyshev's bound, stated below in the versions used. The preprint itself says (Remark 3.6) that nothing here controls the width of the cluster interval; for $c=2$ the accepted Lean proof recorded on the problem page collapses the interval to the point $2$, and for $c\ne2$ the limit $V(cx)/V(x)\to c$ remains open.

Definitions and imported inputs

For real $x$ let $T(x)$ be the set of integers $n$ with $1\le n\le x$ and $n=\varphi(m)$ for some integer $m\ge1$, and $V(x)=|T(x)|$. Fix a real $c>1$ and put $R_c(x)=V(cx)/V(x)$ for real $x\ge1$. Write $\log_kx$ for the $k$-fold iterated natural logarithm, and $\pi(y)$ for the number of primes up to $y$.

Elementary facts about $V$. $V$ is nondecreasing and integer-valued, so $R_c(x)\ge1$. For $0\le h\le1$ the interval $(x,x+h]$ contains at most one integer and $(cx,c(x+h)]$ at most $\lceil ch\rceil\le\lceil c\rceil$ integers, so

$$0\le V(x+h)-V(x)\le1,\qquad 0\le V(c(x+h))-V(cx)\le\lceil c\rceil .$$

Chebyshev's lower bound. There is an absolute constant $c_0>0$ with $\pi(y)\ge c_0\,y/\log y$ for all real $y\ge2$. Since $p\mapsto p-1=\varphi(p)$ injects the primes $p\le x+1$ into $T(x)$,

$$V(x)\ge\pi(x+1)\ge c_0\,\frac{x}{\log x}\qquad(x\ge2).$$

Ford's Theorem 1. Theorem 1 of Ford (1998) (the held paper's §1.1, stated on its card): there are constants $C=0.8178\ldots$ and $D=2.1769\ldots$ such that, for all large $x$,

$$V(x)=\frac{x}{\log x}\exp\bigl\{\Psi(x)+O(1)\bigr\},\qquad \Psi(x)=C(\log_3x-\log_4x)^2+D\log_3x-\bigl(D+\tfrac12-2C\bigr)\log_4x .$$

The held Ford PDF is the author's later corrected text, not the 1998 journal print: its Remark after Theorem 3 (p. 3) says that the proof of Theorem 3 in the journal paper, cited there as [14] (p. 42), contains an error and gives a corrected proof with a weaker estimate, and its metadata date is 2012. The preprint's reference [5] is the arXiv revision (arXiv:1104.3264v2, 2013); its display (2) on p. 3 agrees with the statement above, which is the held text's Theorem 1 (p. 2). The 1998 journal print is not held, and whether the held file's bytes coincide with the arXiv posting was not checked. Only the following consequence is used: with $M(x)=(x/\log x)e^{\Psi(x)}$ there are $K>0$ and $x_1$ such that

$$V(x)=M(x)\,e^{E(x)},\qquad|E(x)|\le K\qquad(x\ge x_1). \tag{F}$$

The values of $C$ and $D$ play no role.

Statement

For every fixed real $c>1$:

1. $\liminf_{n\to\infty}|V(cn)/V(n)-c|=0$, the limit inferior taken over the integers $n$.
2. More precisely, for every function $L(X)\to\infty$ there is a function $\omega(X)\to0$, depending on $c$ and $L$, such that for all large $X$ some integer $n\in[X,cXL(X)]$ satisfies $|R_c(n)-c|\le\omega(X)$.
3. The set $\mathcal C_c$ of subsequential limits of $R_c(n)$ as $n\to\infty$ through the integers is the closed interval $[\alpha_c,\beta_c]$ with $\alpha_c=\liminf_nR_c(n)$ and $\beta_c=\limsup_nR_c(n)$, and $\alpha_c\le c\le\beta_c$; the same set is obtained as $x\to\infty$ through the reals.

Consequently exactly one of the following holds: $R_c(x)\to c$; or $\mathcal C_c$ is a nondegenerate closed interval containing $c$, so that $R_c$ has continuum many limit points. Nothing below bounds $\beta_c-\alpha_c$.

Proof

Step 1: the logarithmic profile (Lemma 2.1)

For real $t$ large enough that $c^t\ge x_1$ and $\log_4(c^t)$ is defined, put $u=t\log c=\log(c^t)$, $\psi_c(t)=\Psi(c^t)$ and

$$\eta_c(t)=-\log(t\log c)+\psi_c(t).$$

Then $\log M(c^t)=\log(c^t)-\log\log(c^t)+\Psi(c^t)=t\log c+\eta_c(t)$, so (F) reads

$$\log V(c^t)=t\log c+\eta_c(t)+E(c^t),\qquad|E(c^t)|\le K . \tag{5}$$

Now $\log_3(c^t)=\log\log u$ and $\log_4(c^t)=\log\log\log u$, and $du/dt=\log c$, so

$$\frac{d}{dt}\log_3(c^t)=\frac{\log c}{u\log u}=\frac{1}{t\log u},\qquad \frac{d}{dt}\log_4(c^t)=\frac{1}{t\log u\,\log\log u}.$$

For $u\ge e^e$ both derivatives lie in $(0,1/(t\log u)]$ and $0\le\log_4(c^t)\le\log_3(c^t)=\log\log u$. Differentiating $\psi_c$,

$$\psi_c'(t)=2C\bigl(\log_3(c^t)-\log_4(c^t)\bigr) \Bigl(\frac{d}{dt}\log_3(c^t)-\frac{d}{dt}\log_4(c^t)\Bigr) +D\,\frac{d}{dt}\log_3(c^t)-\bigl(D+\tfrac12-2C\bigr)\frac{d}{dt}\log_4(c^t),$$

so

$$|\psi_c'(t)|\le\frac{2C\log\log u+D+|D+\tfrac12-2C|}{t\log u}\le\frac{K'}{t}, \qquad K'=2C+D+|D+\tfrac12-2C|,$$

because $\log\log u\le\log u$. Together with the derivative $-1/t$ of $-\log(t\log c)$ this gives $|\eta_c'(t)|\le K_1/t$ for $t\ge t_0(c)$, with $K_1=K'+1$. Integrating over $[N,N+H]$ for $N\ge t_0(c)$ and $H\ge1$,

$$|\eta_c(N+H)-\eta_c(N)|\le K_1\int_N^{N+H}\frac{dt}{t} =K_1\log\Bigl(1+\frac HN\Bigr).$$

Step 2: the geometric block mean (Theorem 2.2)

Let $N\ge t_0(c)$ be an integer and $H\ge1$. The product of consecutive quotients telescopes exactly:

$$\prod_{j=N}^{N+H-1}R_c(c^j)=\prod_{j=N}^{N+H-1}\frac{V(c^{j+1})}{V(c^j)} =\frac{V(c^{N+H})}{V(c^N)} .$$

Taking logarithms and applying (5) at $t=N+H$ and $t=N$,

$$\sum_{j=N}^{N+H-1}\log R_c(c^j)=H\log c+\eta_c(N+H)-\eta_c(N)+E(c^{N+H})-E(c^N),$$

and Step 1 bounds the last four terms by $K_1\log(1+H/N)+2K$. Hence, with $\Delta(N,H)=(1+\log(1+H/N))/H$ and $K_2=\max(K_1,2K)$,

$$\Bigl|\frac1H\sum_{j=N}^{N+H-1}\log R_c(c^j)-\log c\Bigr|\le K_2\,\Delta(N,H). \tag{6}$$

Step 3: boundedness and unit-interval variation (Lemma 3.1)

First, $R_c$ is bounded. By (F), for $x\ge x_1$,

$$R_c(x)=\frac{M(cx)}{M(x)}\,e^{E(cx)-E(x)}\le e^{2K}\frac{M(cx)}{M(x)},\qquad \frac{M(cx)}{M(x)}=c\cdot\frac{\log x}{\log(cx)}\cdot e^{\Psi(cx)-\Psi(x)} .$$

Here $\log x/\log(cx)\to1$, and writing $x=c^t$, Step 1 gives $|\Psi(cx)-\Psi(x)|=|\psi_c(t+1)-\psi_c(t)|\le K'/t\to0$. So $M(cx)/M(x)\to c$, and there are $K_c$ and $x_2(c)$ with

$$1\le R_c(x)\le K_c\qquad(x\ge x_2(c)).$$

Now let $x\ge\max(x_2(c),2)$ and $0\le h\le1$. Put $A=V(cx)$, $B=V(x)$, $\delta=V(c(x+h))-V(cx)\in[0,\lceil c\rceil]$ and $\epsilon=V(x+h)-V(x)\in[0,1]$. Then

$$R_c(x+h)-R_c(x)=\frac{A+\delta}{B+\epsilon}-\frac AB =\frac{B\delta-A\epsilon}{B(B+\epsilon)},$$

so, using $B+\epsilon\ge B$, $\delta\le\lceil c\rceil$, $\epsilon\le1$ and $A/B\le K_c$,

$$|R_c(x+h)-R_c(x)|\le\frac{\delta}{B}+\frac{A\epsilon}{B^2} \le\frac{\lceil c\rceil+K_c}{V(x)}\le K_3\,\frac{\log x}{x}, \tag{7}$$

with $K_3=(\lceil c\rceil+K_c)/c_0$ by Chebyshev's bound. The right side tends to $0$.

Step 4: the integer block mean (Proposition 3.2)

Put $n_j=\lceil c^j\rceil$, so $0\le n_j-c^j<1$, and take an integer $N\ge t_0(c)$ so large that $c^N\ge\max(x_2(c),2)$ and $c^N\ge e$. By (7) with $x=c^j$ and $h=n_j-c^j$,

$$|R_c(n_j)-R_c(c^j)|\le K_3\,\frac{\log(c^j)}{c^j}=K_3\log c\cdot\frac{j}{c^j} \qquad(j\ge N).$$

Both quotients are at least $1$, and $|\log a-\log b|\le|a-b|$ for $a,b\ge1$ by the mean value theorem, so the same bound holds for the logarithms. Summing,

$$\Bigl|\sum_{j=N}^{N+H-1}\bigl(\log R_c(n_j)-\log R_c(c^j)\bigr)\Bigr| \le K_3\log c\sum_{j\ge N}\frac{j}{c^j}\le K_4\,\frac{N}{c^N},$$

because

$$\sum_{j\ge N}\frac{j}{c^j}=c^{-N}\sum_{i\ge0}\frac{N+i}{c^i} \le Nc^{-N}\sum_{i\ge0}\frac{1+i}{c^i}=\frac{c^2}{(c-1)^2}\cdot\frac{N}{c^N}.$$

Combining with (6),

$$\Bigl|\frac1H\sum_{j=N}^{N+H-1}\log R_c(n_j)-\log c\Bigr| \le K_2\,\Delta(N,H)+K_4\,\frac{N}{Hc^N}. \tag{P}$$

Step 5: a near-hit in every sampled block (Theorem 3.3, display (8))

Let $s_j=R_c(n_j)$ for $N\le j\le N+H-1$ and let $S=(\prod_js_j)^{1/H}$ be their geometric mean, so that $\log S-\log c$ is the left side of (P). Since $\Delta(N,H)\le1+\log2$ for all $N,H\ge1$ (the function $H\mapsto(1+\log(1+H))/H$ is decreasing) and $N/c^N$ is bounded, the left side of (P) is bounded by a constant $K_5$, and $|e^v-1|\le e^{|v|}|v|$ gives

$$|S-c|\le c\,e^{K_5}\Bigl(K_2\,\Delta(N,H)+K_4\,\frac{N}{Hc^N}\Bigr).$$

If some $s_j$ equals $c$ the block contains an exact hit. Otherwise one of three cases holds.

Every $s_j>c$. A geometric mean is at least the minimum, so $\min_j|s_j-c|=\min_js_j-c\le S-c$.

Every $s_j<c$. A geometric mean is at most the maximum, so $\min_j|s_j-c|=c-\max_js_j\le c-S$.

Some $s_j<c$ and some $s_k>c$. Then there are consecutive indices $i,i+1$ in the block with $s_i$ and $s_{i+1}$ on opposite sides of $c$ (walk from $j$ toward $k$ and stop at the first change of side). Say $s_i<c<s_{i+1}$; the other case is symmetric. Since $c^{i+1}-c^i\ge(c-1)c^N>1$ for large $N$, $n_i<n_{i+1}$. Let $m^\ast$ be the largest integer in $[n_i,n_{i+1})$ with $R_c(m^\ast)<c$; then $R_c(m^\ast+1)\ge c$, so $c$ lies between $R_c(m^\ast)$ and $R_c(m^\ast+1)$, and by (7) with $h=1$,

$$|R_c(m^\ast)-c|\le R_c(m^\ast+1)-R_c(m^\ast) \le K_3\,\frac{\log m^\ast}{m^\ast} \le K_3\,\frac{\log(c^N)}{c^N}=K_3\log c\cdot\frac{N}{c^N},$$

because $t\mapsto(\log t)/t$ is decreasing for $t\ge e$ and $m^\ast\ge n_N\ge c^N\ge e$.

In every case, using $N/(Hc^N)\le N/c^N$, there is a constant $C_c$ with

$$\min_{\substack{x\in\mathbb N\\ n_N\le x\le n_{N+H-1}}}|R_c(x)-c| \le C_c\Bigl(\Delta(N,H)+\frac{N}{c^N}\Bigr) \tag{8}$$

for all large $N$ and all $H\ge1$.

Step 6: near-hits in every growing multiplicative window (display (9))

Let $L(X)\to\infty$ and set $N=\lceil\log_cX\rceil$ and $H=\lfloor\log_cL(X)\rfloor$. For large $X$, $L(X)\ge c$ gives $H\ge1$, and $N\to\infty$, $H\to\infty$. The sampled block lies in $[X,cXL(X)]$: $n_N\ge c^N\ge X$, while $c^N<c^{\log_cX+1}=cX$ and $c^{H-1}\le L(X)/c$ give

$$n_{N+H-1}<c^{N+H-1}+1<cX\cdot\frac{L(X)}{c}+1=XL(X)+1\le cXL(X)$$

once $(c-1)XL(X)\ge1$. Hence by (8),

$$\min_{\substack{x\in\mathbb N\\ X\le x\le cXL(X)}}|R_c(x)-c| \le\omega(X):=C_c\Bigl(\Delta(N,H)+\frac{N}{c^N}\Bigr)\longrightarrow0, \tag{9}$$

since $\Delta(N,H)\le(1+\log(1+H))/H\to0$ as $H\to\infty$ and $N/c^N\to0$ as $N\to\infty$. This is clause 2 of the statement. Clause 1 follows by taking, say, $L(X)=X$: for each large integer $k$ there is an integer $n_k\in[k,ck^2]$ with $|R_c(n_k)-c|\le\omega(k)\to0$, and $n_k\to\infty$.

Step 7: the cluster interval (Theorem 3.4 and Corollary 3.5)

By Step 3, $1\le R_c(n)\le K_c$ for large integers $n$, so $\alpha_c\le\beta_c$ are finite, both are subsequential limits, and $\mathcal C_c$ is a closed subset of $[\alpha_c,\beta_c]$. By (7) with $h=1$, the adjacent variation $|R_c(n+1)-R_c(n)|\le K_3(\log n)/n$ tends to $0$.

Let $\alpha_c<y<\beta_c$ and let $n_0\ge\max(x_2(c),3)$ be given. Since $\alpha_c<y$ there is $n_1>n_0$ with $R_c(n_1)<y$, and since $\beta_c>y$ there is $n_2>n_0$ with $R_c(n_2)>y$. If $n_1<n_2$, let $n$ be the largest integer in $[n_1,n_2)$ with $R_c(n)<y$, so that $R_c(n+1)\ge y$; if $n_2<n_1$, let $n$ be the largest integer in $[n_2,n_1)$ with $R_c(n)>y$, so that $R_c(n+1)\le y$. Either way $n>n_0$ and $y$ lies between $R_c(n)$ and $R_c(n+1)$, so

$$|R_c(n)-y|\le|R_c(n+1)-R_c(n)|\le K_3\,\frac{\log n_0}{n_0}.$$

Letting $n_0$ run through an increasing sequence produces integers $n^{(k)}\to\infty$ with $R_c(n^{(k)})\to y$, so $y\in\mathcal C_c$. Hence $(\alpha_c,\beta_c)\subseteq\mathcal C_c\subseteq[\alpha_c,\beta_c]$, and closedness gives $\mathcal C_c=[\alpha_c,\beta_c]$. By clause 1, $c\in\mathcal C_c$, that is, $\alpha_c\le c\le\beta_c$.

For the real variable, let $x\ge1$ be real, $n=\lfloor x\rfloor$ and $h=x-n\in[0,1)$; (7) gives $|R_c(x)-R_c(n)|\le K_3(\log n)/n\to0$. So a sequence of reals $x_k\to\infty$ has $R_c(x_k)\to y$ exactly when $R_c(\lfloor x_k\rfloor)\to y$, and the sets of subsequential limits over the reals and over the integers coincide. This is clause 3.

Finally, exactly one of $\alpha_c=\beta_c$ and $\alpha_c<\beta_c$ holds. In the first case $R_c(n)$ converges to the common value, which is $c$ since $c\in[\alpha_c,\beta_c]$, and the real-variable statement gives $R_c(x)\to c$. In the second, $\mathcal C_c$ is a nondegenerate interval containing $c$, hence uncountable. This is Corollary 3.5 and completes the proof of Theorem 1.1.

The dyadic renewal identity (Proposition 6.1), for $c=2$

The set of totient values is closed under doubling: if $v=\varphi(m)$ and $m$ is even, write $m=2^am'$ with $a\ge1$ and $m'$ odd, so that $\varphi(2m)=2^a\varphi(m')=2\cdot2^{a-1}\varphi(m')=2\varphi(m)$; if $m$ is odd, $\varphi(4m)=\varphi(4)\varphi(m)=2\varphi(m)$. Call a totient value $v$ dyadically primitive if $v/2$ is not a totient value (this includes $v=1$, the only odd value), and let $P(x)$ be the number of dyadically primitive values in $[1,x]$.

Every totient value $v$ is $2^kb$ for exactly one pair $(k,b)$ with $k\ge0$ and $b$ dyadically primitive. Existence: halve $v$ while the result is a totient value; the process stops after $k\le\log_2v$ steps at a primitive $b$. Uniqueness: if $2^kb=2^{k'}b'$ with $b,b'$ primitive and $k>k'$, then $b'=2^{k-k'}b$ and $b'/2=2^{k-k'-1}b$ is a totient value by closure under doubling, contradicting the primitivity of $b'$; so $k=k'$ and $b=b'$. Counting the values in $[1,x]$ by $k$,

$$V(x)=\sum_{k\ge0}P\bigl(x/2^k\bigr),$$

a finite sum since $P(z)=0$ for $z<1$. Replacing $x$ by $2x$ and shifting the index,

$$V(2x)-V(x)=P(2x)+\sum_{k\ge1}P\bigl(2x/2^k\bigr)-\sum_{k\ge0}P\bigl(x/2^k\bigr) =P(2x).$$

So the number of totient values in $(x,2x]$ equals the number of dyadically primitive values up to $2x$; the preprint uses the identity for its entropy recursion.

What is not reconstructed

The matched quotient $(V(c^2x)-V(cx))/(V(cx)-V(x))$ and its cluster interval (Theorem 4.1, Lemma 4.2, Corollary 4.3), which use Ford's Theorem 4 ($V(cx)-V(x)\asymp_cV(x)$); the block-energy identity and the local second-moment criterion (Proposition 5.1, Lemma 5.2, Theorem 5.3), a sufficient condition for the limit that the preprint says no known estimate verifies; the entropy recursion along dyadic orbits (Theorem 6.2, Corollary 6.3); the log-periodic model (Proposition 7.1), whose stated role is that bounded-factor asymptotics, monotonicity, unit jumps and telescoping do not by themselves force the limit, so the argument above cannot be pushed to $R_c(x)\to c$ without arithmetic input; and the segmented computation of §8 reporting $V(10^{10})=1{,}311{,}179{,}363$, unverified here. None of these bears on the proof above.