975.lean`

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

7 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 975

References:

erdosproblems.com/975
[Va39] van der Corput, J. G., Une in'egalit'e{} relative au nombre des diviseurs. Nederl. Akad. Wetensch., Proc. (1939), 547--553.
[Er52b] Erd"os, P., On the sum {$\sum^x_{k=1} d(f(k))$}. J. London Math. Soc. (1952), 7--15.
[Ho63] Hooley, Christopher, On the number of divisors of a quadratic polynomial. Acta Math. (1963), 97--114.
[Mc95] McKee, James, On the average number of divisors of quadratic polynomials. Math. Proc. Cambridge Philos. Soc. (1995), 389--392.
[Mc97] McKee, James, A note on the number of divisors of quadratic polynomials. (1997), 275--281.
[Mc99] McKee, James, The average number of divisors of an irreducible quadratic polynomial. Math. Proc. Cambridge Philos. Soc. (1999), 17--22.
[T] T. Tao, Erdos' divisor bound, https://terrytao.wordpress.com/2011/07/23/erdos-divisor-bound/

Imports (1)

FormalConjecturesUtil

def Erdos975.Erdos975Sum Sum of $\tau(f(n))$ from 0 to ⌊x⌋ for a polynomial $f \in \mathbb{Z}[X]$ line 47 source
Source documentation
Sum of $\tau(f(n))$ from 0 to ⌊x⌋ for a polynomial $f \in \mathbb{Z}[X]$.
Here $\tau$ is the divisor counting function, which is σ 0 in mathlib. Also, for simplicity, we use Nat.floor to convert rational values to natural numbers, instead of dealing with negative values.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Int, Real
Proof uses: Finset.sum, Nat, Finset.Iic, Nat.floor, ArithmeticFunction, ArithmeticFunction.sigma, OfNat.ofNat, Polynomial.eval
29 instances, coercions and auxiliary constants left out.
```
noncomputable def Erdos975Sum (f : ℤ[X]) (x : ℝ) : ℝ :=
…
```
theorem Erdos975.erdos_975 For an irreducible polynomial sorry line 58 source
Source documentation
For an irreducible polynomial $f \in \mathbb{Z}[x]$ with $f(n) \ge 1$ for sufficiently large $n$, does there exists a constant $c = c(f) > 0$ such that $\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x$?
Note that it is unclear whether the polynomial should have integer coefficients or merely be integer-valued. We assume the former.

depends on sorry what this means
Statement uses: Iff, True, Polynomial, Int, Ne, Nat, Polynomial.natDegree, OfNat.ofNat, Irreducible, Filter.Eventually, LE.le, Polynomial.eval, Filter.atTop, Exists, Real, And, GT.gt, Filter.Tendsto, HDiv.hDiv, Erdos975.Erdos975Sum and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_975 : answer(sorry) ↔
    ∀ f : ℤ[X], f.natDegree ≠ 0 → Irreducible f → (∀ᶠ n in atTop, 1 ≤ f.eval n) →
    ∃ c > (0 : ℝ), Tendsto (fun x ↦ Erdos975Sum f x / (x * log x)) atTop (𝓝 c) := by
…
```
Mentions: c, Erdos975.Erdos975Sum
theorem Erdos975.erdos_975.variants.upper_bound The correctness of the growth rate is shown in [Va39] (lower bound) and [Er52b] (upper bound) sorry line 67 source
Source documentation
The correctness of the growth rate is shown in [Va39] (lower bound) and [Er52b] (upper bound).

depends on sorry what this means
Statement uses: Polynomial, Int, Irreducible, Filter.Eventually, LE.le, OfNat.ofNat, Polynomial.eval, Filter.atTop, Asymptotics.IsBigO, Real, Erdos975.Erdos975Sum, HMul.hMul, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_975.variants.upper_bound (f : ℤ[X]) (hf : Irreducible f)
    (hf_pos : ∀ᶠ n in atTop, 1 ≤ f.eval n) : Erdos975Sum f =O[atTop] (fun x ↦ x * log x) := by
…
```
Mentions: Erdos975.Erdos975Sum
theorem Erdos975.erdos_975.variants.lower_bound Lower bound for the growth rate of Erdos975Sum, shown in [Va39] sorry line 73 source
Source documentation
Lower bound for the growth rate of Erdos975Sum, shown in [Va39].

depends on sorry what this means
Statement uses: Polynomial, Int, Irreducible, Ne, Nat, Polynomial.natDegree, OfNat.ofNat, Filter.Eventually, LE.le, Polynomial.eval, Filter.atTop, Asymptotics.IsBigO, Real, HMul.hMul, Real.log, Erdos975.Erdos975Sum
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_975.variants.lower_bound (f : ℤ[X]) (hf : Irreducible f) (hfdeg : f.natDegree ≠ 0)
    (hf_pos : ∀ᶠ n in atTop, 1 ≤ f.eval n) :
    (fun x ↦ x * log x) =O[atTop] Erdos975Sum f := by
…
```
Mentions: Erdos975.Erdos975Sum
theorem Erdos975.erdos_975.variants.quadratic When $f$ is an irreducible quadratic polynomial, the question is answered first by Hooley [Ho63] sorry line 89 source
Source documentation
When $f$ is an irreducible quadratic polynomial, the question is answered first by Hooley [Ho63]. More compact expression of the constant in terms of Hurwitz class numbers (when $a = 1$) is given by McKey in [Mc95], [Mc97], [Mc99].
TODO: formalize Hurwitz class numbers and the expression of the constant in terms of them.

depends on sorry what this means
Statement uses: Polynomial, Int, Irreducible, Filter.Eventually, Nat, LE.le, OfNat.ofNat, Polynomial.eval, Filter.atTop, Eq, WithBot, Polynomial.degree, Real, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, And and 7 more
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_975.variants.quadratic (f : ℤ[X]) (hf : Irreducible f)
    (hf_pos : ∀ᶠ n : ℕ in atTop, 1 ≤ f.eval ↑n) (hf_degree : f.degree = 2) (c : ℝ) :
    c = answer(sorry) → 0 < c ∧ Tendsto (fun x ↦ Erdos975Sum f x / (x * log x)) atTop (𝓝 c) := by
…
```
Mentions: c, Erdos975.Erdos975Sum
theorem Erdos975.erdos_975.variants.n2_plus_1_strong More concrete example for $f(n) = n^2 + 1$, where the asymptote is $\sum_{n \le x} \tau(n^2 + 1) \sim \frac{3}{\pi} x \log x + O(x)$ sorry line 99 source
Source documentation
More concrete example for $f(n) = n^2 + 1$, where the asymptote is $\sum_{n \le x} \tau(n^2 + 1) \sim \frac{3}{\pi} x \log x + O(x)$. See Tao's blog [T].

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Real, Filter.atTop, HSub.hSub, Erdos975.Erdos975Sum, HAdd.hAdd, Polynomial, Int, HPow.hPow, Nat, Polynomial.X, OfNat.ofNat, HMul.hMul, HDiv.hDiv, Real.pi, Real.log, id
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_975.variants.n2_plus_1_strong :
    (fun x ↦ Erdos975Sum (X ^ 2 + 1) x - (3 / π) * x * log x) =O[atTop] id := by
…
```
Mentions: Erdos975.Erdos975Sum
theorem Erdos975.erdos_975.variants.n2_plus_1 Asymptotics for Erdos975Sum with $f(X) = X^2 + 1$ sorry line 105 source
Source documentation
Asymptotics for Erdos975Sum with $f(X) = X^2 + 1$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Tendsto, HDiv.hDiv, Erdos975.Erdos975Sum, HAdd.hAdd, Polynomial, Int, HPow.hPow, Nat, Polynomial.X, HMul.hMul, Real.log, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_975.variants.n2_plus_1 :
    ∃ c > (0 : ℝ), Tendsto (fun x ↦ Erdos975Sum (X ^ 2 + 1) x / (x * log x)) atTop (𝓝 c) := by
…
```
Mentions: c, Erdos975.Erdos975Sum