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Math-808-A2 · revision r1 · on Math-808

gpt pro 5.2 attempt on Math-808

No claim recorded · LLM-generated · gpt pro 5.2 ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. FORMAL RESTATEMENT
  2. QUICK LITERATURE/CONTEXT CHECK
  3. ATTACK PLAN
  4. WORK
  5. Fast reality check (exact computation for small $x$)
  6. Lemma 416.1 (Parity of totients)
  7. Lemma 416.2 (Primes give many distinct totients)
  8. VERIFICATION
  9. FINAL

FORMAL RESTATEMENT

Definitions.
  • $\mathbb{N}=\{1,2,3,\dots\}$.
  • Euler's totient function $\varphi:\mathbb{N}\to\mathbb{N}$ is \[ \varphi(m)=\#\{1\le a\le m : \gcd(a,m)=1\}. \]
  • For real $x\ge 0$, define \[ V(x)=\#\{n\in\mathbb{N}: n\le x \ \text{and}\ \exists m\in\mathbb{N}\ \text{with}\ \varphi(m)=n\}. \] (Equivalently, $V(x)$ is the number of distinct totients $\le x$.)

Questions.

  • $Q1$ Does $\displaystyle \lim_{x\to\infty} \frac{V(2x)}{V(x)}=2$?
  • $Q2$ Does $V(x)$ admit an asymptotic formula as $x\to\infty$?
Edge cases. For $0\le x<1$ one has $V(x)=0$. For $1\le x<2$ one has $V(x)=1$ since $\varphi(1)=\varphi(2)=1$.

QUICK LITERATURE/CONTEXT CHECK

The problem statement (as given in the extracted file) records progressively sharper information on $V(x)$: Pillai proved $V(x)=o(x)$, Erdős proved $V(x)=x(\log x)^{-1+o(1)}$, Maier--Pomerance proved \[ V(x)=\frac{x}{\log x}\exp\big((C+o(1))(\log\log\log x)^2\big), \] and Ford proved a two-sided estimate of the shape \[ V(x)\asymp \frac{x}{\log x}\exp\Big( C_1(\log\log\log x-\log\log\log\log x)^2+ C_2\log\log\log x- C_3\log\log\log\log x\Big) \] with explicit constants $C_i>0$. The remaining issue is that these do not yield an asymptotic constant, nor do they settle whether $V(2x)/V(x)\to 2$.

ATTACK PLAN

Proof-track ideas (for $V(2x)/V(x)\to 2$).
  • Use the best-known description of $V(x)$ as $\frac{x}{\log x}$ times a "slowly varying" factor, and attempt to show that factor is asymptotically stable under $x\mapsto 2x$.
  • Translate the question into a statement about how many new totients appear between $x$ and $2x$ (i.e. $V(2x)-V(x)$) and seek matching upper/lower bounds.
Disproof-track ideas.
  • Try to exhibit sequences $x_j$ for which $V(2x_j)/V(x_j)$ oscillates, by forcing unusually many/few totients near the endpoints.
  • Look for "rigidity" obstructions: for instance, totients are mostly even and have strong multiplicative constraints; see if those could force $V(2x)/V(x)$ away from $2$.
Given the deep context already cited in the statement, I focus below on rigorous elementary bounds and sanity-check computations; I do not attempt to reproduce the cited deep estimates.

WORK

Fast reality check (exact computation for small $x$)

Using the inequality from Problem #417 below (Lemma 417.1), any totient value $\le x$ occurs as $\varphi(m)$ for some $m\le 2x^2$. Thus $V(x)$ can be computed exactly for moderate $x$ by enumerating $\varphi(m)$ for $m\le 2x^2$ and counting distinct values $\le x$.

Exact values obtained (via a totient sieve) are: 

 x    V(x)   V(2x)   V(2x)/V(x)
 10     6      10      1.6666666667
 20    10      17      1.7000000000
 50    21      38      1.8095238095
 100   38      72      1.8947368421
 200   72     131      1.8194444444
 500  158     291      1.8417721519
 1000 291     543      1.8659793814
These ratios are far from conclusive but show $V(2x)/V(x)<2$ for these $x$.

Lemma 416.1 (Parity of totients)

Lemma. If $m>2$ then $\varphi(m)$ is even. Consequently, if $n>1$ and $\varphi(m)=n$ for some $m$, then $n$ is even. Therefore, for all real $x\ge 0$, \[ V(x)\le 1+\big\lfloor \tfrac{x}{2}\big\rfloor. \]

Proof. Fix $m>2$. Let \[ R=\{a\in\{1,2,\dots,m\}: \gcd(a,m)=1\}. \] Then $\#R=\varphi(m)$ by definition. Define a map $T:R\to R$ by $T(a)=m-a$. If $\gcd(a,m)=1$ then $\gcd(m-a,m)=\gcd(a,m)=1$, so $T$ maps $R$ to itself. Also $T$ is an involution: $T(T(a))=a$. If $T(a)=a$ then $m-a=a$, so $m=2a$ and in particular $m$ is even and $a=m/2$. But then $\gcd(a,m)=\gcd(m/2,m)=m/2>1$ (since $m>2$), contradicting $a\in R$. Hence $T$ has no fixed points on $R$, so it partitions $R$ into disjoint $2$-element orbits. Therefore $\#R$ is even, i.e. $\varphi(m)$ is even.

Now suppose $\varphi(m)=n>1$. Then $m\ne 1,2$ because $\varphi(1)=\varphi(2)=1$, hence $m>2$ and the first part implies $n$ is even. Thus the set of totients $\le x$ is contained in $\{1\}\cup\{2,4,6,\dots,2\lfloor x/2\rfloor\}$. This set has size $1+\lfloor x/2\rfloor$, so $V(x)$ is at most this size. $\square$

Lemma 416.2 (Primes give many distinct totients)

Lemma. For all real $x\ge 1$, \[ V(x)\ge \pi(x+1), \] where $\pi(y)$ is the number of primes $\le y$.

Proof. Let $p$ be any prime with $p\le x+1$. Then $\varphi(p)=p-1$ (since the reduced residues modulo a prime are exactly $1,2,\dots,p-1$). Hence $p-1$ is a totient value and $p-1\le x$. If $p\ne q$ are distinct primes then $p-1\ne q-1$, so the totients $\{p-1: p\le x+1\text{ prime}\}$ are all distinct. Therefore there are at least $\pi(x+1)$ distinct totients $\le x$, i.e. $V(x)\ge \pi(x+1)$. $\square$

VERIFICATION

  • Lemma 416.1 explicitly excludes the exceptional cases $m=1,2$ where $\varphi(m)=1$ is odd; the proof checks that the involution $a\mapsto m-a$ has no fixed point because $a=m/2$ would not be coprime to $m$.
  • Lemma 416.2 uses only the identity $\varphi(p)=p-1$ for primes $p$, and injectivity of $p\mapsto p-1$.
  • The computation of $V(x)$ used the bound $m\le 2x^2$ for totients $\le x$ proved as Lemma 417.1 below; this is independently verifiable.

FINAL

UNRESOLVED
  1. (i) Strongest proved partial result here. Elementary bounds \[ \pi(x+1)\le V(x)\le 1+\lfloor x/2\rfloor \] for all $x\ge 1$ (Lemmas 416.1--416.2), together with exact computed values of $V(x)$ for $x\le 1000$ (table above).
  2. (ii) First gap (crisp). Prove or disprove \[ \lim_{x\to\infty}\frac{V(2x)}{V(x)}=2. \] Equivalently, show that for every $\varepsilon>0$ there exists $X$ such that for all $x\ge X$, $\big|V(2x)-2V(x)\big|\le \varepsilon V(x)$; or exhibit a sequence $x_j\to\infty$ along which the ratio fails to approach $2$.
  3. (iii) Top 3 next moves.
    • Try to upgrade the two-sided estimates for $V(x)$ stated in the problem (in particular the explicit exponent structure in Ford's estimate) to a true asymptotic, or at least show the multiplicative correction factor is "slowly varying" enough to imply $V(2x)/V(x)\to 2$.
    • Compute $V(x)$ for much larger $x$ (e.g. up to $10^7$ or $10^8$) using a totient sieve plus the bound $m\le 2x^2$, and numerically examine $V(2x)/V(x)$ to search for systematic drift or oscillation.
    • Study the increment $V(2x)-V(x)$ by estimating the number of totients landing in $(x,2x]$; any matching upper and lower bounds of order $V(x)$ would be decisive.
  4. (iv) What a minimal counterexample would likely look like. If $V(2x)/V(x)$ fails to converge to $2$, one expects a sequence $x_j$ where the "local density" of totients near $x_j$ is atypically high or low, producing persistent oscillation in $V(2x_j)-2V(x_j)$. A minimal counterexample would likely come from constructing intervals with anomalously many (or few) new totient values, rather than from parity-type constraints (already accounted for in Lemma 416.1).