The counting problem.

Since \(\varphi(p)=p-1\) for primes \(p\), the prime number theorem gives \(V(x)\ge\pi(x+1)\sim x/\log x\). Pillai proved that the totient values have density zero (Pillai 1929). Erdős used the normal number of prime factors of \(p-1\) to sharpen the upper bound to \(V(x)\ll_\varepsilon x/(\log x)^{1-\varepsilon}\) for every \(\varepsilon>0\) (Erdős 1935, sec. 2). His later count of distinct values \(\varphi(pr)\) with \(p,r\) prime and \(pr\le x\) gives the lower bound \((1+o(1))x\log_2x/\log x\) (Erdős 1945, 542–43). Erdős and Hall obtained a much larger factor \(\exp\{a(\log_3x)^2\}\) for every \(0<a<1/\log16\) (Erdős and Hall 1976, 1–3). Pomerance brought the upper bound onto the same logarithmic scale (Pomerance 1986, Equation (1.4)). Maier and Pomerance then determined the leading constant (Maier and Pomerance 1988, sec. 1): \[V(x)=\frac{x}{\log x} \exp\{(C_0+o(1))(\log_3x)^2\},\qquad C_0=0.8178146\ldots.\]

In 1998, Ford refined this to an estimate with only a bounded multiplicative uncertainty (Ford 1998, Theorem 1): \[ V(x)=\frac{x}{\log x}\exp\left\{ \begin{aligned} &C_0(\log_3x-\log_4x)^2+D_0\log_3x\\ &\hspace{12pt}-(D_0+\tfrac12-2C_0)\log_4x+O(1) \end{aligned}\right\}, \tag{1}\] where \(D_0=2.1769687\ldots\); both constants are specified in the notation of Section 2 below. Ford also described the typical prime factorizations of preimages (Ford 1998, Theorems 10 and 16), which supply the structural starting point of our proof. All numbered references to his distribution paper use the revised arXiv version of 14 July 2013.

Fixed dilation.

Erdős and Hall asked whether \[\frac{V(cx)}{V(x)}\longrightarrow c \qquad(c>1\text{ fixed})\] in the final paragraph of (Erdős and Hall 1976, 3). Erdős repeated the question in (Erdős 1979, 80). Ford proved that \(V(cx)-V(x)\asymp_c V(x)\) for every fixed \(c>1\) (Ford 1998, Theorem 4). This controls the number of values in a fixed multiplicative interval, but neither it nor (1) determines the limiting ratio.

We obtain an explicit asymptotic equivalent and prove the fixed-scale limit, thereby answering this question of Erdős and Hall positively. The coefficient in the equivalent retains finite arithmetic information. It is defined in Section 2 by bounded prime and integer sums, inclusion–exclusion, and a uniformly convergent limit; its definition does not use \(V\).

The source of the coefficient.

Order the prime factors of a typical preimage from largest to smallest, and separate a long prefix from a shorter tail. Ford’s normal-structure estimates put the double logarithms of the large primes in a simplex. Most of that simplex can be counted by volume. Near its last coordinates the primes remain discrete, however, and several tail factorizations may give the same totient. We retain these tails exactly and take the volume of the union of the regions that they permit. Inclusion–exclusion of this finite union produces the arithmetic coefficient.

Erdős’s semiprime argument already controls collisions between products of shifted primes (Erdős 1945, Lemma 1). The quantitative shifted-prime normality and layered comparison used here were developed by Maier and Pomerance (Maier and Pomerance 1988, secs. 2–3) and Ford (Ford 1998, sec. 5). Here the comparison is proved under the hypotheses needed for the long prefix, with explicit costs for recovering the tuple data inside the residual factors. The resulting control of all ordered prefix collisions, combined with the exact finite-tail union, converts the volume into a count of distinct values. Counting the largest prime with common endpoint parameters first gives regular variation. Identifying the resulting mass with the arithmetic coefficient then gives the explicit equivalent. The errors are uniform in the phase; neither conclusion assumes continuity of the coefficient.

The prefix uniqueness used here is not uniqueness of the entire preimage: different arithmetic tails may still give the same value. A separate result on extreme complete fibers gives, for every fixed \(\varepsilon>0\), infinitely many positive integers \(v\) with \(\#\{n\ge1:\varphi(n)=v\}>v^{1-\varepsilon}\) (OpenAI 2026, Theorem 1.1). That result concerns large individual fibers, rather than the asymptotic number of distinct values, and is not an input to the present proof.

Least preimages.

For \(v\in\mathcal V\), let \[\ell(v)=\min\{n\ge1:\varphi(n)=v\}.\] For each fixed positive integer \(k\), the same passage of Erdős (Erdős 1979, 80) asks about the number of values \(v\le x\) for which \(kx<\ell(v)\le(k+1)x\). Section 7 gives a weighted asymptotic and proves that this count has order \(V(x)\) precisely when some totient \(d\) satisfies \(\ell(d)>kd\); otherwise the count vanishes identically. The positive alternative holds for \(k=1,2\). We do not determine which other integers \(k\) satisfy this seed condition. In particular, the separate question whether \(\ell(d)/d\) is unbounded, posed in (Erdős 1995, sec. I.9), remains unresolved here.

The scale

For \(j\ge1\), put \[ a_j=(j+1)\log(j+1)-j\log j-1 =\int_j^{j+1}\log t\,\mathrm dt. \tag{2}\] These numbers are positive. The increasing function \(\sum_{j\ge1}a_jz^j\) goes from zero to infinity on \(0<z<1\), so there is a unique \(\rho\in(0,1)\) such that \[\sum_{j\ge1}a_j\rho^j=1.\] These coefficients and this root occur in Maier and Pomerance’s asymptotic estimate (Maier and Pomerance 1988, sec. 1). Define \[ \lambda=\log(1/\rho),\qquad \gamma=\left(\sum_{j\ge1}ja_j\rho^j\right)^{-1},\qquad g_0=1,\quad g_j=\sum_{d=1}^j a_dg_{j-d}. \tag{3}\] Our \(\gamma\) is the constant denoted by \(\lambda\) in Ford’s recurrence estimates (Ford 1998, sec. 3). Set \[C_0=\frac1{2\lambda},\qquad D_0=\frac{1+\log(\lambda/\gamma)}{\lambda}-\frac12;\] these are the constants in the classical estimate (1). For the scale used in our theorem, put \[ B=\log_2x,\qquad m=\left\lfloor\frac{\log B-\log_2B}{\lambda}\right\rfloor, \qquad \theta=\frac{\log B-\log_2B}{\lambda}-m, \tag{4}\] so that \(0\le\theta<1\), and write \[ G_j=\frac{B^j}{j!\prod_{i=1}^j g_i}\quad(0\le j\le m),\qquad G_0=1. \tag{5}\] The factor \(xG_m/\log x\) is Ford’s counting scale. The following theorem resolves its bounded multiplicative uncertainty by an explicitly constructed function of the phase \(\theta\).

The first argument of \(A\) records a weight; the main theorem uses the constant weight \(1\). Its arithmetic construction below does not use \(V\). The fixed-scale conclusion has a separate mechanism: it compares the same representation count at two endpoints. No regularity of \(s\mapsto A(1;s)\) is asserted or needed.

The arithmetic coefficient

The tail index runs in the opposite direction from the ordered prime index. In the proof we write a preimage as \(p_0p_1\cdots p_La\), with the primes decreasing, and use two cuts \[R=m-H,\qquad L=m-P,\qquad P=\lfloor\log\log H\rfloor.\] The long prefix ends at \(p_R\), whereas the exact arithmetic tail contains \(p_{R+1},\ldots,p_L\) and \(a\). Thus the notation below is \(Q_h=p_{m-h}\): its indices run from \(P\) to \(H-1\), and \(Q_P\) is the smallest retained prime. For fixed \(H\), these tail primes are bounded independently of \(x\). We first let \(x\) tend to infinity with \(H\) fixed, and only afterwards let \(H\) tend to infinity. The following definition uses the phase \(s\) directly, without reference to \(x\).

Fix \(s\in[0,1)\) and put \(\alpha_s=\lambda\mathrm e^{\lambda s}\). Let \(H\) be a sufficiently large positive integer and \(P=\lfloor\log\log H\rfloor\). A tail witness is a choice \[\eta=((Q_h)_{P\le h<H},a)\] of primes \(Q_h\) and a positive integer \(a\) with the following properties. Writing \(v_h=\log_2Q_h\), we require \[ \begin{aligned} 0.9\alpha_s h\rho^{-h}&\le v_h\le1.1\alpha_s h\rho^{-h} &&(P\le h<H),\\ \sum_{l=P}^{h-1}a_{h-l}v_l &\le(1+10^{-4}\mathrm e^{-h/40})v_h &&(P\le h<H),\\ P^+(a)&\le Q_P,\qquad \log a\le\exp(2\alpha_sP\rho^{-P}). \end{aligned} \tag{6}\] Here \(P^+(a)\) is the largest prime factor of \(a\), with \(P^+(1)=1\); an empty sum is zero. Associate to \(\eta\) the integer \[w(\eta)=a\prod_{h=P}^{H-1}Q_h.\] For \(d\in\mathcal V\), let \(\mathcal W_{H,s}(d)\) be the set of tail witnesses for which \(\varphi(w(\eta))=d\). For \(h\ge H\), define \[ D(h,\eta)=\sum_{l=P}^{H-1}a_{h-l}v_l. \tag{7}\] The set of all possible witnesses, even as \(s\) varies, is finite for each fixed \(H\): \(\alpha_s\) lies between \(\lambda\) and \(\lambda/\rho\), and every prime and integer in (6) is consequently bounded in terms of \(H\). Also \(D(h,\eta)=O_H(\log(h+1))\). Thus the series used below converges absolutely, uniformly over this finite set and over \(s\).

A tail totient must contribute once even when it has several witnesses. The union that enforces this can be described by independent exponential random variables \(E_h\) of mean one, indexed by \(h\ge H\). For each witness put \[ \mathcal E_\eta= \bigcap_{h\ge H} \left\{E_h\ge\frac\gamma{\alpha_s}\rho^hD(h,\eta)\right\}. \tag{8}\] For a function \(f:[1,\infty)\to[0,1]\), the coefficient is \[A_H(f;s)=\rho^{H(H-1)/2} \left(\frac\gamma{\alpha_s}\right)^H \sum_{\substack{d\in\mathcal V\\\mathcal W_{H,s}(d)\ne\varnothing}} \frac{f(\ell(d)/d)}d\, \Pr\left(\bigcup_{\eta\in\mathcal W_{H,s}(d)} \mathcal E_\eta\right).\] An intersection of events requires each \(E_h\) to exceed the largest of the corresponding thresholds. Independence and absolute convergence of their sum therefore turn finite inclusion–exclusion into the explicit expression \[ \begin{split} A_H(f;s)={}&\rho^{H(H-1)/2} \left(\frac{\gamma}{\alpha_s}\right)^H \sum_{\substack{d\in\mathcal V\\\mathcal W_{H,s}(d)\ne\varnothing}} \frac{f(\ell(d)/d)}d\\[-2pt] &\quad\times \sum_{\varnothing\ne T\subseteq\mathcal W_{H,s}(d)} (-1)^{|T|-1} \exp\left\{-\frac{\gamma}{\alpha_s} \sum_{h=H}^{\infty}\rho^h \max_{\eta\in T}D(h,\eta)\right\}. \end{split} \tag{9}\] For \(f=1\), the factor involving \(\ell\) is simply \(1\): no least-preimage search is part of the main coefficient. For other weights, any witness with totient \(d\) supplies the finite search bound \(\ell(d)\le w(\eta)\). The probability expression also shows directly that \(A_H(f;s)\ge0\). Section 6 identifies these probabilities with limiting normalized prefix volumes: \(1/d\) comes from counting the largest prime, and the prefactor is the limiting ratio of the prefix volume to \(G_m\).

When the limit exists, write \[A(f;s)=\lim_{H\to\infty}A_H(f;s).\] Each approximant uses finite arithmetic data and an absolutely convergent series, not an unspecified counting constant. Theorem 1 justifies the limit for \(f=1\); the next result does so for the weights needed to count least preimages. Convergence for arbitrary \(f\) is not required.

Least-preimage intervals

For a positive integer \(k\), define \[ \begin{aligned} N_k(x)&=\#\{v\in\mathcal V:v\le x,\ kx<\ell(v)\le(k+1)x\},\\ f_k(r)&=\min\{1,(k+1)/r\}-\min\{1,k/r\}. \end{aligned} \tag{10}\] The function \(f_k\) is nonnegative and bounded by \(1\).

This theorem does not determine which other integers \(k\) satisfy the first alternative. Whether the first alternative holds for every positive integer \(k\) is equivalent to the unboundedness of \(\ell(d)/d\); the weighted formula does not settle this question.

Proof strategy

Two parts of the argument ensure that this construction counts values once. First, outside an exceptional set, every preimage has the required form. Second, collisions between different long prefixes have negligible total multiplicity. For the latter we use the layered shifted-prime method of Maier and Pomerance (Maier and Pomerance 1988, sec. 3), in the form developed by Ford, and adapt his comparison (Ford 1998, Lemma 5.1) to a different set of hypotheses, proving the necessary uniform bound in Section 4. The number of layers grows, but each sieve application concerns only two or three linear forms. The comparison and the subsequent control of tuple multiplicities are the ingredients that turn a volume estimate into an asymptotic for distinct values.

The end of the continuous prefix and the end of the retained prime list are different cuts. Their separation makes the exceptional-set estimates small enough to absorb the smooth tail. Section 3 obtains the required preimages directly from Ford’s Theorems 10 and 16 and develops the volume estimates. Section 5 proves prefix uniqueness and transfers tuple counts to distinct values. Section 6 counts the largest prime using common endpoint parameters, obtaining fixed dilation before identifying the arithmetic limit. Finally, Section 7 weights that prime count to locate least preimages and proves the positive and zero alternatives.

Structure estimates from Ford

Ford’s order estimate and fundamental-simplex estimates give \[ V(x)\asymp \frac{x}{\log x}G_m, \tag{14}\] and his recurrence estimate gives \[ g_j=\gamma\rho^{-j}+O(1),\qquad g_j\asymp\rho^{-j}. \tag{15}\] See (Ford 1998, Theorem 1, Lemmas 3.4 and 3.7, and Corollary 3.5). Ford denotes our constant \(\gamma\) by \(\lambda\). In particular, for \(0\le j\le m\), \[ \frac{G_j}{G_m} =\prod_{i=j+1}^m\frac{ig_i}{B} \le C^{m-j}\rho^{(m-j)(m-j-1)/2}\ll1. \tag{16}\]

We shall use two quantitative structure results of Ford. Order the prime factors of a preimage \(n\) with repetitions as \(p_0\ge p_1\ge\cdots\). For each fixed \(0<\eta\le1/8\), his Theorem 10, together with the missing-prime exception in Theorem 16, implies \[ \#\left\{v\le x:\begin{array}{l} v\in\mathcal V,\ \text{some preimage has}\\ |\log_2p_i/b_i-1|>\eta\text{ for some }1\le i\le m-P \end{array}\right\} \ll_\eta V(x)\mathrm e^{-c_\eta P}. \tag{17}\] Here and below a missing indicated prime is a failure of the condition. For completeness, the summation giving (17) is short. For \(i\le3\eta m\), Theorem 10(b) contributes at most \(Cm\mathrm e^{-\eta m/13}V(x)\). For \(i>3\eta m\), Theorem 10(a) contributes at most \[C V(x)\frac{i}{\eta m} \exp\left\{-\frac{\eta^2m(m-i)}{4i}\right\} \le \frac{CV(x)}{\eta}\mathrm e^{-\eta^2(m-i)/4}.\] Summing over \(m-i\ge P\) proves the assertion for sufficiently large \(x\) when the indicated primes exist. A missing \(p_L\) is included in the \(O(V(x)\mathrm e^{-P^2/4})\) exceptional set of Theorem 16, which is absorbed in the displayed bound. The convention preceding Ford’s Theorem 10 counts values with some failing preimage; hence the complement of this set controls every preimage.

Ford’s Theorem 16, with its parameter \(\Psi=P\), states that all but \(O(V(x)\mathrm e^{-P^2/4})\) values have every preimage satisfying \[ \log_2p_L>2,\qquad \sum_{r=i+1}^L a_{r-i}u_r\le\xi_i u_i\quad(0\le i<L), \qquad u_0=B,\quad u_i=\log_2p_i\ (i\ge1). \tag{18}\] Its final adjacent-coordinate inequality has coefficient \(1\) in place of \(a_1<1\); weakening that inequality gives (18).

Write \(\Omega(n,U,T)\) for the number of prime factors of \(n\) belonging to \((U,T]\), counted with multiplicity, and put \(\Omega(n)=\Omega(n,1,n)\), including \(\Omega(1)=0\).

We use Definition 1 in the 2013 revision (Ford 1998). Its Lemma 2.6 gives, uniformly for \(z\ge3\), \[ \#\{p\le z:p\text{ is not }S\text{-normal}\} \ll \frac{z}{\log z}(\log_2z)^5(\log S)^{-1/6}. \tag{19}\] The function \(\Omega\) in these statements counts prime factors with multiplicity.

Basic witnesses and the coverage target

We now specify the representations that these structure estimates will provide. The simplex notation records the inequalities on their prime coordinates; the bounds on the last factor will make the arithmetic tail finite for fixed \(H\).

For integers \(1\le j\le m\), \(u_0=B\), and \(\beta_i\ge1\) for \(0\le i<j\), put \(\beta_j=1\) and define the simplex \[S_j(\boldsymbol\beta)= \left\{(u_1,\ldots,u_j): s_i:=\beta_i u_i-\sum_{r=i+1}^j a_{r-i}u_r\ge0 \quad(0\le i\le j)\right\}.\] The terminal inequality is \(u_j\ge0\). For \(j=0\), the simplex consists of the empty tuple and has volume \(G_0=1\).

For our parameter range the prime bands are disjoint and decreasing; also \(p_0>p_1\). Thus the displayed prime prefix is strictly decreasing. Repetitions of \(p_L\) inside \(a\) are allowed.

We prove the proposition after the volume and smooth-cofactor estimates below. Those estimates also prepare a second task: we shall retain the shortened data \((p_0,\ldots,p_R,\varphi(w))\), then delete those that fail the strict inequalities and normality conditions needed for the collision argument. Coverage concerns values; this later deletion must count the distinct shortened tuples, including all tuples that represent the same value.

Simplex volume and concentration

The coordinate scaling follows Ford’s fundamental-simplex method (Ford 1998, Lemma 3.2 and Corollary 3.3). Our last adjacent coefficient is \(a_1\) rather than \(1\), so the exact volume is slightly different from his; we derive it from the slacks below. The recurrence estimate (15) is Ford’s Lemma 3.7; his Remark 1 credits its streamlined proof to Lau’s solution of a recurrence problem (Ford and Lau 2000).

The next estimates turn sums over the allowed prime coordinates into simplex volumes. We also bound the mass near a coordinate boundary or a simplex face, so that removing these regions has a controlled cost.

The following concentration estimate will be used in Section 6 to remove the prefix bands after replacing prime sums by volumes. Write \(z_+=\max\{z,0\}\).

Prime boxes and smooth cofactors

The use of reciprocal prime sums and enlarged coordinate boxes follows Ford’s Lemma 3.1 and the thickening arguments of his Section 3 (Ford 1998). We record the quantitative bounds for our simplexes, including the boundary errors needed for the tail unions.

We use unit boxes in the coordinates \(\log_2p_i\). Mertens’ theorem gives \[ \sum_{b\le\log_2p<b+1}\frac1{p-1}=1+O(\mathrm e^{-b}) \qquad(b\ge2). \tag{22}\] The replacement of \(1/p\) by \(1/(p-1)\) changes the sum by \(O(\exp(-\exp b))\). When the lower endpoints grow geometrically backwards through the coordinates, multiplying (22) costs only a bounded factor; if their sum of errors is small, it is the relative product error.

Combining Lemmas 8 and 9, the shell estimates remain valid after summing a smooth cofactor with weight \(1/\varphi(a)\), at a cost of at most \(K_H\). In particular their weighted mass is bounded by \[ C K_HG_m\left\{H^{-2}+ \sum_{h\ge H}(h\mathrm e^{-h/40}+h^3\rho^h)\right\}. \tag{27}\] The \(H^{-2}\) term is needed only for condition (iii).

Coverage of the values

Tuples and their discarded part

For \(t=x\) or \(t=x/c\), where \(c>1\) is fixed, keep all parameters formed from \(x\). Let \(\mathcal T(t)\) be the set of distinct tuples \[\tau=(p_0,\ldots,p_R,d)\] admitting at least one basic witness with \[d=\varphi(w),\qquad w=p_{R+1}\cdots p_La,\qquad d\prod_{i=0}^R(p_i-1)\le t.\] Each tuple represents a totient, because \(w<p_R\) makes its tail coprime to its prefix. Several witnesses may give the same tuple; \(\mathcal T(t)\) counts that tuple once.

We next remove tuples for which the comparison argument might fail. For each \(0\le i\le R\), put \(h=h_i\) and \[J_i=\lceil30\log h\rceil,\qquad S_i=\exp\exp(b_i^{1/3}),\qquad z_i=\exp\exp(0.7b_{i+J_i}),\] and, for a basic witness, put \[D_i=\varphi(p_{i+J_i+1}\cdots p_La).\] For large \(H\) these indices exist: \(J_i<h_i/2\) and \(i+J_i<L\). We call a tuple good if every basic witness giving it satisfies, for every \(0\le i\le R\), \[ \begin{gathered} \sum_{r=i+1}^L a_{r-i}u_r\le(1-h_i^{-4})u_i,\\ p_i,\ldots,p_{i+J_i}\text{ are }S_i\text{-normal},\\ (p_i-1)\cdots(p_{i+J_i}-1) \text{ is squarefree on primes exceeding }z_i,\\ \Omega(D_i)\le b_i/h_i^8 . \end{gathered} \tag{28}\] As before, the first condition uses \(u_0=B\). Denote the good tuples by \(\mathcal G(t)\), and put \(\mathcal D(t)=\mathcal T(t)\setminus\mathcal G(t)\).

The two cuts have now served different purposes. The cut at \(L\) keeps the smooth-cofactor cost at \(K_H=H^{o(1)}\); the earlier cut at \(R\) supplies the summable \(H^{-2}\) loss and the much stronger savings in the remaining discards. Proposition 5 controls exceptional values, while Proposition 10 controls exceptional tuples. Neither yet asserts uniqueness of a representation. That is the purpose of the comparison and collision arguments that follow.

A uniform sieve for two or three forms

The comparison estimate

For a positive integer \(n\) and a real number \(Z\), its part supported above \(Z\) means \(\prod_{p^a\parallel n,\ p>Z}p^a\). We use \(S\)-normality as defined in Definition 3.

The key term in the bound below is the exponent of \(\log y\). In its application, a strict simplex inequality makes the weighted cutoff sum plus its error smaller than \(1\) by a controlled amount. The exponent is consequently smaller than \(-1\); that extra saving absorbs the other factors, even after recovering the tuple data hidden in the smooth remainder. We need an estimate for ordered pairs with multiplicity: a bound only on the number of values admitting a collision would not control how many tuples lie over those values.

Counting pairs with the same value

The alignment of largest factors and cancellation follow the argument around (5.34)–(5.37) in Ford’s Section 5 (Ford 1998). Here we count all ordered pairs of distinct tuples, including the data inside both residual factors. We first cancel the common prefix and align the large factors in the remaining shifts. After applying the comparison estimate, we recover the tuple coordinates inside the residual factors and then restore the common prefix.

From tuples to values and least preimages

The following elementary lemma keeps track of exceptional tuples as well as exceptional values. This is needed because a small set of values could otherwise carry many representations.

The universal quantifier on preimages in Proposition 5 is used only after the counting argument, to obtain the common-prefix statement and (56). A single basic preimage would give the upper bound for \(\ell(v)\) but would not give the reverse inequality.

Summing over the largest prime

Let \(\mathcal D_H(x)\) be the set of distinct data \((d,p_1,\ldots,p_R)\) for which there are primes \(p_{R+1},\ldots,p_L\) and a positive integer \(a\) satisfying the basic conditions (20) with \(p_0\) omitted, and \[d=\varphi(w),\qquad w=p_{R+1}\cdots p_La.\] The convention \(u_0=B\) is retained in these conditions. Thus the inequality with index zero is a condition on the displayed data and their witness, independent of the eventual choice of \(p_0\). For any function \(f:[1,\infty)\to[0,1]\), define \[ M_f(x;H)= \sum_{(d,p_1,\ldots,p_R)\in\mathcal D_H(x)} \frac{f(\ell(d)/d)}{d\prod_{i=1}^R(p_i-1)}. \tag{57}\] Each datum is included once, regardless of its number of witnesses. The function \(f=1\) makes the least-preimage ratio irrelevant to this definition.

We use \(K_H=\exp(CP\rho^{-P})=H^{o(1)}\), allowing the absolute constant \(C\) to increase in upper bounds. Summing over witnesses, the totient inequality \[ \frac{1}{\varphi(p_{R+1}\cdots p_La)} \le \frac{1}{\varphi(a)} \prod_{i=R+1}^L\frac{1}{p_i-1} \tag{58}\] and Lemma 8 give \[ 0\le M_f(x;H)\le M_1(x;H)\ll K_HG_m. \tag{59}\] Indeed, the reciprocal sum for the full \(L\) prime coordinates is \(O(G_m)\) by (23), while the unrestricted reciprocal sum for the allowed smooth cofactor is \(O(K_H)\) by (26). The inequality in (58), rather than an equality, also covers the possibility that \(p_L\) divides \(a\).

The common mass in Proposition 17 already gives the fixed-scaling conclusion. Its proof does not require identifying that mass with the arithmetic coefficient.

A finite tail and a continuous prefix

The phase-dependent witness set in (9) changes with the phase, but for fixed \(H\) all its members belong to one finite set. This observation will permit uniform limits without any continuity assumption.

For \(s=\theta(x)\) and \(\eta\in\mathcal W_{H,s}(d)\), define the region \(\mathcal K_\eta\subset\mathbb R^R\) by \(u_0=B\) and \[ u_i-\sum_{r=i+1}^Ra_{r-i}u_r\ge D(m-i,\eta) \qquad(0\le i\le R). \tag{62}\] These inequalities imply \(u_i\ge0\). The coordinates \(u_i\) here are real; the coarse prime bands have been removed. We write \(\operatorname{Vol}_R\) for \(R\)-dimensional Lebesgue measure.

The arithmetic approximation

Uniform convergence and the asymptotic equivalent

The preceding approximation holds for every bounded nonnegative weight. Convergence as \(H\) grows requires a comparison quantity that does not depend on \(H\). For \(f=1\) this quantity is the normalized count of totients. Section 7 will supply a different comparison quantity for each \(f_k\).

Counting the interval for the largest prime

The additive form is necessary at this stage. For example, when \(r=k\) and \(q<1\), one has \(f_k(r)=0\) but \(f_k(r/q)>0\). Inequality (73) controls this boundary without asserting a relative approximation to a zero term.

Apply Lemma 22 with the function \[C_{f_k}(x)=\frac{N_k(x)\log x}{xG_m},\] which is independent of \(H\). Lemma 23 provides its required comparison with \(M_{f_k}/G_m\). We obtain the uniform convergence of \(A_H(f_k;s)\) on \(0\leq s<1\) and \[ N_k(x)=\frac{xG_m}{\log x} \bigl(A(f_k;\theta)+o(1)\bigr). \tag{74}\] It remains to distinguish a coefficient bounded away from zero from an identically zero count.

Seed propagation and bounded ratios

The positivity argument needs two distinct facts. A seed with \(\ell(d)/d>k\) must produce a positive proportion of values whose ratios stay away from \(k\). These ratios must also remain bounded on a positive proportion of values, since \(f_k(r)\to0\) as \(r\to\infty\). The next two lemmas supply these facts.

Preservation of a complete inverse fiber under multiplication by a prime already appears in Erdős’s proof of Theorem 4 (Erdős 1958). Ford’s construction gives such preservation on a positive proportion of the distinct-totient scale. A later simultaneous two-seed form appears in Pollack, Pomerance and Treviño (Pollack et al. 2013, Lemma 4.1), again using Ford’s construction. We use the following single-seed consequence of Ford’s proof.

We also need to keep these ratios in a bounded interval. This follows from the existence of a bounded arithmetic tail and does not require uniqueness of the prefix.

The positive and zero alternatives

Suppose first that there is a totient \(d_*\) with \(\ell(d_*)>kd_*\). Choose \(\delta>0\) such that \(\ell(d_*)/d_*>k+2\delta\). Lemma 24 gives a fixed \(\eta>0\) such that at least \(\eta V(x)\) values satisfy \(\ell(v)/v>k+2\delta\) for every sufficiently large \(x\). Apply Lemma 25 with \(\varepsilon=\eta/4\). It supplies a fixed \(C\) for which at least \(3\eta V(x)/4\) values satisfy \[ k+2\delta<\frac{\ell(v)}v\leq C. \tag{75}\] The numbers \(\eta,\delta,C\) are fixed before the cut \(H\) is allowed to grow.

For every sufficiently large \(H\), and then sufficiently large \(x\), Proposition 16 removes at most \(\eta V(x)/4\) of these values. For each remaining value its unique tuple satisfies \[\frac{\ell(v)}v=\frac{\ell(d)}d \prod_{i=0}^R\frac{p_i}{p_i-1}.\] The product is \(1+O(\epsilon_H)+o_{x\to\infty;H}(1)\), uniformly in the tuple. Increasing \(H\) and then \(x\), we may bound it by \((k+2\delta)/(k+\delta)\). Thus at least \(\eta V(x)/2\) distinct tuples have their tail ratio in the fixed interval \[I=[k+\delta,C].\]

Let \(M_{\mathbf 1_I}\) denote the mass (57) with weight \(\mathbf 1_I(r)\). The ordinary prime number theorem, counting all admissible largest primes for data whose tail ratio lies in \(I\), bounds the number of such tuples above by \[\bigl(1+o_{x\to\infty;H}(1)\bigr) \frac{x}{\log x}M_{\mathbf 1_I}(x;H).\] Comparison with \(\eta V(x)/2\), followed by (14), gives \[M_{\mathbf 1_I}(x;H)\geq c_1G_m\] for all sufficiently large \(H\) and then sufficiently large \(x\), where \(c_1>0\) is independent of \(H\). Since \[\inf_{r\in I}f_k(r) \geq\min\left\{\frac{\delta}{k+\delta},\frac1C\right\}>0,\] we conclude that \[ M_{f_k}(x;H)\geq c_2G_m \tag{76}\] with \(c_2>0\) independent of sufficiently large \(H\).

We transfer this lower bound first to the normalized count \(C_{f_k}(x)=N_k(x)\log x/(xG_m)\), which does not depend on \(H\). Fix a sufficiently large \(H\) such that the error \(\epsilon_{H,k}\) in (71) is less than \(c_2/4\), and then take \(x\) large enough that its remaining error is less than \(c_2/4\). Equations (71) and (76) give \(C_{f_k}(x)\geq c_2/2\) for every sufficiently large \(x\). This count is also bounded above, since \(N_k(x)\leq V(x)\) and (14) holds. The bounds clause of Lemma 22 therefore gives \[\inf_{0\leq s<1}A(f_k;s)\geq c_2/2>0.\] Therefore the additive formula (74) is multiplicative in this case. Its positive lower bound, the inequality \(N_k(x)\leq V(x)\), and (14) give \(N_k(x)\asymp_k V(x)\).

If instead \(\ell(d)\leq kd\) for every totient \(d\), then \(\ell(v)\leq kv\leq kx\) for every \(v\leq x\). Hence \(N_k(x)=0\) for every \(x\). Every weight \(f_k(\ell(d)/d)\) in the finite arithmetic formula is also zero, so \(A_H(f_k;s)=A(f_k;s)=0\). This proves the dichotomy in Theorem 2. Its final assertion is established next.