50.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 50

References:

erdosproblems.com/50
[Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.
[Sch38] Schoenberg, I. J. "On asymptotic distributions of arithmetical functions." Transactions of the American Mathematical Society 39.2 (1936): 315-330.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def Erdos50.IsDistributionOfPhiRatio Documented declaration: IsDistributionOfPhiRatio line 43 source
Source documentation
A function $f : \mathbb{R} \to \mathbb{R}$ is the asymptotic distribution function of the values of $\varphi(n)/n$ if for all $c \in [0, 1]$, the natural density of $\{n : \varphi(n) < cn\}$ exists and equals $f(c)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Membership.mem, Set, Set.Icc, OfNat.ofNat, Set.HasDensity, Nat, Set.ofPred, LT.lt, Nat.totient, HMul.hMul, Set.univ
15 instances, coercions and auxiliary constants left out.
```
def IsDistributionOfPhiRatio (f : ℝ → ℝ) : Prop :=
…
```
def Erdos50.IsPurelySingularOn Documented declaration: IsPurelySingularOn line 51 source
Source documentation
A function $f : \mathbb{R} \to \mathbb{R}$ is purely singular (or singular continuous) on a set $s \subseteq \mathbb{R}$ if it is continuous on $s$ and its derivative within $s$ equals zero almost everywhere on $s$ with respect to Lebesgue measure.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Set
Proof uses: And, ContinuousOn, Filter.Eventually, Eq, derivWithin, OfNat.ofNat, MeasureTheory.ae, MeasureTheory.Measure, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume
18 instances, coercions and auxiliary constants left out.
```
def IsPurelySingularOn (f : ℝ → ℝ) (s : Set ℝ) : Prop :=
…
```
theorem Erdos50.erdos_50_schoenberg Schoenberg [Sch38] proved that the asymptotic distribution function of $\varphi(n)/n$ exists sorry line 61 source
Source documentation
Schoenberg [Sch38] proved that the asymptotic distribution function of $\varphi(n)/n$ exists. That is, for any $c \in [0, 1]$, the proportion of integers $n \le N$ satisfying $\varphi(n)/n < c$ approaches a limit as $N \to \infty$. This limit function is the cumulative distribution function of the values of $\varphi(n)/n$.

depends on sorry what this means
Statement uses: Exists, Real, Erdos50.IsDistributionOfPhiRatio
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_50_schoenberg : ∃ f : ℝ → ℝ, IsDistributionOfPhiRatio f := by
…
```
Mentions: Erdos50.IsDistributionOfPhiRatio
theorem Erdos50.erdos_50_singular Documented declaration: erdos_50_singular sorry line 69 source
Source documentation
Erdős [Er95] proved that the distribution function of $\varphi(n)/n$ is purely singular on $[0, 1]$: it is continuous there, but its derivative is zero almost everywhere on $[0, 1]$.

depends on sorry what this means
Statement uses: Real, Erdos50.IsDistributionOfPhiRatio, Erdos50.IsPurelySingularOn, Set.Icc, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_50_singular (f : ℝ → ℝ) (hf : IsDistributionOfPhiRatio f) :
    IsPurelySingularOn f (Icc 0 1) := by
…
```
Mentions: Erdos50.IsDistributionOfPhiRatio, Erdos50.IsPurelySingularOn
theorem Erdos50.erdos_50 Let $f$ be the asymptotic distribution function of $\varphi(n)/n$, so that for each $c \in [0,1]$, $f(c)$ is the natural density of $\{n : \varphi(n) < cn\}$ sorry line 79 source
Source documentation
Let $f$ be the asymptotic distribution function of $\varphi(n)/n$, so that for each $c \in [0,1]$, $f(c)$ is the natural density of $\{n : \varphi(n) < cn\}$. Is it true that there is no $x$ such that the derivative $f'(x)$ exists and is positive?

depends on sorry what this means
Statement uses: Iff, True, Real, Erdos50.IsDistributionOfPhiRatio, Not, Exists, And, Membership.mem, Set, Set.Icc, OfNat.ofNat, GT.gt, HasDerivWithinAt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
91 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_50 : answer(sorry) ↔ ∀ᵉ (f : ℝ → ℝ) (hf : IsDistributionOfPhiRatio f),
    ¬∃ x ∈ Icc (0 : ℝ) 1, ∃ y > 0, HasDerivWithinAt f y (Icc 0 1) x := by
…
```
Mentions: Erdos50.IsDistributionOfPhiRatio