{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1243",
  "content_hash": null,
  "slug": "erdos-problem-50-attempt-resolution-route",
  "type": "attempt",
  "title": "Resolve the stated acceptance condition",
  "summary": "Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\\varphi$ denote Euler's totient function. A function $f : \\mathbb{R} \\to \\mathbb{R}$ is called the asymptotic distribution function of $\\varphi(n)/n$ if for every $c \\in [0, 1]$, the set $\\{n \\in \\mathbb{N} : \\varphi(n) < cn\\}$ has a natural density, and this density equals $f(c)$. Determine whether the following holds: for every function $f : \\mathbb{R} \\to \\mathbb{R}$ that is the asymptotic distribution function of $\\varphi(n)/n$, there do not exist $x \\in [0, 1]$ and $y > 0$ such that $f$ has derivative $y$ at $x$ within the interval $[0, 1]$.",
  "relevance": "For Non-existence of positive derivatives for the distribution function of Euler's totient ratio, record erdos-problem-50-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\\varphi$ denote Euler's totient function.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\\varphi$ denote Euler's totient function. A function $f : \\mathbb{R} \\to \\mathbb{R}$ is called the asymptotic distribution function of $\\varphi(n)/n$ if for every $c \\in [0, 1]$, the set $\\{n \\in \\mathbb{N} : \\varphi(n) < cn\\}$ has a natural density, and this density equals $f(c)$. Determine whether the following holds: for every function $f : \\mathbb{R} \\to \\mathbb{R}$ that is the asymptotic distribution function of $\\varphi(n)/n$, there do not exist $x \\in [0, 1]$ and $y > 0$ such that $f$ has derivative $y$ at $x$ within the interval $[0, 1]$. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.",
  "status": "open_strategy",
  "evidence_grade": "self_reported",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://www.erdosproblems.com/50",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/50",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R1244",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-50",
      "title": "erdos problem 50",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}