RiemannZetaValues.lean`

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Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

8 declarations, 8 contain sorry, 0 declared as axioms

Particular values of the Riemann zeta function

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

Imported by 2 files

theorem RiemannZetaValues.irrational_five sorry line 41 source
depends on sorry what this means
Statement uses: Exists, Real, And, Irrational, Eq, Complex, riemannZeta, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem RiemannZetaValues.irrational_seven $\zeta(7)$ is irrational sorry line 48 source
Source documentation
$\zeta(7)$ is irrational.

depends on sorry what this means
Statement uses: Exists, Real, And, Irrational, Eq, Complex, riemannZeta, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_seven : ∃ x, Irrational x ∧ riemannZeta 7 = x := by
…
```
theorem RiemannZetaValues.irrational_nine $\zeta(9)$ is irrational sorry line 55 source
Source documentation
$\zeta(9)$ is irrational.

depends on sorry what this means
Statement uses: Exists, Real, And, Irrational, Eq, Complex, riemannZeta, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_nine : ∃ x, Irrational x ∧ riemannZeta 9 = x := by
…
```
theorem RiemannZetaValues.irrational_eleven $\zeta(11)$ is irrational sorry line 62 source
Source documentation
$\zeta(11)$ is irrational.

depends on sorry what this means
Statement uses: Exists, Real, And, Irrational, Eq, Complex, riemannZeta, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_eleven : ∃ x, Irrational x ∧ riemannZeta 11 = x := by
…
```
theorem RiemannZetaValues.irrational_odd $\zeta(2n + 1)$ is irrational for any $n\in\mathbb{N}^{+}$ sorry line 69 source
Source documentation
$\zeta(2n + 1)$ is irrational for any $n\in\mathbb{N}^{+}$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Exists, Real, And, Irrational, Eq, Complex, riemannZeta, HAdd.hAdd, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_odd (n : ℕ) (hn : 0 < n) :
    ∃ x, Irrational x ∧ riemannZeta (2 * n + 1) = x := by
…
```
theorem RiemannZetaValues.irrational_three $\zeta(3)$ is irrational sorry line 79 source
Source documentation
$\zeta(3)$ is irrational.
[Ap79] Apéry, R. (1979). Irrationalité de ζ(2) et ζ(3). Astérisque. 61: 11–13.

depends on sorry what this means
Statement uses: Exists, Real, And, Irrational, Eq, Complex, riemannZeta, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_three : ∃ x, Irrational x ∧ riemannZeta 3 = x := by
…
```
theorem RiemannZetaValues.infinite_irrational_at_odd There are infinitely many $\zeta(2n + 1)$, $n \in \mathbb{N}$, that are irrational sorry line 88 source
Source documentation
There are infinitely many $\zeta(2n + 1)$, $n \in \mathbb{N}$, that are irrational.
[Ri00] Rivoal, T. (2000). La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. Comptes Rendus de l'Académie des Sciences, Série I. 331 (4): 267–270.

depends on sorry what this means
Statement uses: Set.Infinite, Nat, Set.ofPred, Exists, Real, And, Irrational, Eq, Complex, riemannZeta, HAdd.hAdd, HMul.hMul, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinite_irrational_at_odd :
    { n : ℕ | ∃ x, Irrational x ∧ riemannZeta (2 * n + 1) = x }.Infinite := by
…
```
theorem RiemannZetaValues.exists_irrational_of_five_seven_nine_eleven At least one of $\zeta(5), \zeta(7), \zeta(9)$ or $\zeta(11)$ is irrational sorry line 98 source
Source documentation
At least one of $\zeta(5), \zeta(7), \zeta(9)$ or $\zeta(11)$ is irrational.
[Zu01] W. Zudilin (2001). One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational. Russ. Math. Surv. 56 (4): 774–776.

depends on sorry what this means
Statement uses: Set.Nonempty, Complex, Inter.inter, Set, Insert.insert, OfNat.ofNat, Nat, Singleton.singleton, Set.ofPred, Exists, Real, And, Irrational, Eq, riemannZeta
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_irrational_of_five_seven_nine_eleven :
    {5, 7, 9, 11} ∩ { a | ∃ x, Irrational x ∧ riemannZeta a = x} |>.Nonempty := by
…
```