Lean statement

Finite-order Hecke $L$-function $7/8$ bound

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.SevenEighths.HeckeFamily.LFunction_ne_zero_of_seven_eighths_lt_re

Lean statement file: HeckeSevenEighths.leanShow on the mapFamily 003. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem LFunction_ne_zero_of_seven_eighths_lt_re    (χ : Character) {s : ℂ} (hs : (7 / 8 : ℝ) < s.re)    (hpole : ¬ (χ.residue = 1 ∧ s = 1)) : LFunction χ s ≠ 0 := by sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

definition OAI.SevenEighths.ThetaProduct.product line 32

def product (P Q : WeakFEPair ℂ)    (hPf : ContinuousOn P.f (Ioi 0)) (hPg : ContinuousOn P.g (Ioi 0))    (hQf : ContinuousOn Q.f (Ioi 0)) (hQg : ContinuousOn Q.g (Ioi 0)) :    WeakFEPair ℂ where

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  f := fun x => P.f x * Q.f x  g := fun x => P.g x * Q.g x  k := P.k + Q.k  ε := P.ε * Q.ε  f₀ := P.f₀ * Q.f₀  g₀ := P.g₀ * Q.g₀  hf_int := (hPf.mul hQf).locallyIntegrableOn measurableSet_Ioi  hg_int := (hPg.mul hQg).locallyIntegrableOn measurableSet_Ioi  hk := add_pos P.hk Q.hk  hε := mul_ne_zero P.hε Q.hε  h_feq x hx := by    rw [P.h_feq x hx, Q.h_feq x hx, Real.rpow_add hx]    simp only [smul_eq_mul, Complex.ofReal_mul]    ring  hf_top := rapid_product_sub P.f Q.f P.f₀ Q.f₀ P.hf_top Q.hf_top  hg_top := rapid_product_sub P.g Q.g P.g₀ Q.g₀ P.hg_top Q.hg_top

definition OAI.SevenEighths.ThetaProduct.rescale line 66

def rescale (P : WeakFEPair ℂ)    (hPf : ContinuousOn P.f (Ioi 0)) (hPg : ContinuousOn P.g (Ioi 0))    (u : ℝ) (hu : 0 < u) : WeakFEPair ℂ where

Show the remaining 21 lines

  f := fun x => P.f (u * x)  g := fun x => P.g (u⁻¹ * x)  k := P.k  ε := P.ε * (u ^ (-P.k) : ℝ)  f₀ := P.f₀  g₀ := P.g₀  hf_int := (hPf.comp (by fun_prop) (fun x hx => mul_pos hu hx)).locallyIntegrableOn    measurableSet_Ioi  hg_int := (hPg.comp (by fun_prop) (fun x hx => mul_pos (inv_pos.mpr hu) hx)).locallyIntegrableOn    measurableSet_Ioi  hk := P.hk  hε := mul_ne_zero P.hε (Complex.ofReal_ne_zero.mpr (Real.rpow_pos_of_pos hu _).ne')  h_feq x hx := by    have heq : u * (1 / x) = 1 / (u⁻¹ * x) := by field_simp    rw [heq, P.h_feq _ (mul_pos (inv_pos.mpr hu) hx),      Real.mul_rpow (inv_pos.mpr hu).le hx.le, Real.inv_rpow hu.le,      ← Real.rpow_neg hu.le]    simp only [smul_eq_mul, Complex.ofReal_mul]    ring  hf_top := rapid_rescale P.f P.f₀ P.hf_top hu  hg_top := rapid_rescale P.g P.g₀ P.hg_top (inv_pos.mpr hu)

definition OAI.SevenEighths.ThetaProduct.scaledEvenPair line 111

def scaledEvenPair (a : UnitAddCircle) (u : ℝ) (hu : 0 < u) : WeakFEPair ℂ :=  rescale (HurwitzZeta.hurwitzEvenFEPair a)    (Complex.continuous_ofReal.comp_continuousOn (HurwitzZeta.continuousOn_evenKernel a))    (Complex.continuous_ofReal.comp_continuousOn (HurwitzZeta.continuousOn_cosKernel a)) u hu

definition OAI.SevenEighths.ThetaProduct.rectangularPair line 128

def rectangularPair (a b : UnitAddCircle) (u v : ℝ) (hu : 0 < u) (hv : 0 < v) :    WeakFEPair ℂ :=  product (scaledEvenPair a u hu) (scaledEvenPair b v hv)    (scaledEvenPair_f_continuous a u hu) (scaledEvenPair_g_continuous a u hu)    (scaledEvenPair_f_continuous b v hv) (scaledEvenPair_g_continuous b v hv)

definition OAI.SevenEighths.EisensteinTheta.xShift line 143

def xShift (a b L : ℝ) (e : Fin 2) : ℝ := (2 * a - b - L * e) / (2 * L)

definition OAI.SevenEighths.EisensteinTheta.yShift line 144

def yShift (b L : ℝ) (e : Fin 2) : ℝ := (b + L * e) / (2 * L)

definition OAI.SevenEighths.EisensteinTheta.parityPair line 146

def parityPair (a b L : ℝ) (hL : 0 < L) (e : Fin 2) : WeakFEPair ℂ :=  rectangularPair (xShift a b L e) (yShift b L e) (L ^ 2) (3 * L ^ 2)    (sq_pos_of_pos hL) (mul_pos (by norm_num) (sq_pos_of_pos hL))

definition OAI.SevenEighths.HeckeTheta.finitePair line 161

def finitePair {ι : Type*} [Fintype ι] (P : ι → WeakFEPair ℂ)    (hk : ∀ i, (P i).k = 1) (w : ι → ℂ) : WeakFEPair ℂ where

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  f := ∑ i, w i • (P i).f  g := ∑ i, (w i * (P i).ε) • (P i).g  k := 1  ε := 1  f₀ := ∑ i, w i * (P i).f₀  g₀ := ∑ i, w i * (P i).ε * (P i).g₀  hf_int := locallyIntegrableOn_sum Finset.univ _    (fun i _ => (P i).hf_int.smul (w i))  hg_int := locallyIntegrableOn_sum Finset.univ _    (fun i _ => (P i).hg_int.smul (w i * (P i).ε))  hk := by norm_num  hε := one_ne_zero  h_feq x hx := by    simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, one_mul]    rw [Finset.mul_sum]    apply Finset.sum_congr rfl    intro i _    rw [(P i).h_feq x hx, hk i]    simp only [smul_eq_mul]    ring  hf_top r := by    have h := Asymptotics.IsBigO.fun_sum (s := Finset.univ)      (fun i _ => ((P i).hf_top r).const_mul_left (w i))    simpa only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, ← Finset.sum_sub_distrib,      mul_sub] using h  hg_top r := by    have h := Asymptotics.IsBigO.fun_sum (s := Finset.univ)      (fun i _ => ((P i).hg_top r).const_mul_left (w i * (P i).ε))    simpa only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, ← Finset.sum_sub_distrib,      mul_sub] using h

definition OAI.SevenEighths.HeckeTheta.pair line 194

def pair {N : ℕ} [NeZero N] (w : Fin N × Fin N → ℂ) : WeakFEPair ℂ :=  finitePair (fun p : (Fin N × Fin N) × Fin 2 =>    parityPair p.1.1 p.1.2 N (Nat.cast_pos.mpr (NeZero.pos N)) p.2)    (fun _ => ThetaProduct.rectangularPair_k ..) (fun p => w p.1)

definition OAI.SevenEighths.HeckeTheta.completed line 199

def completed {N : ℕ} [NeZero N] (w : Fin N × Fin N → ℂ) : ℂ → ℂ := (pair w).Λ

definition OAI.SevenEighths.HeckeTheta.latticeL line 201

def latticeL {N : ℕ} [NeZero N] (w : Fin N × Fin N → ℂ) (s : ℂ) : ℂ :=  (Real.pi : ℂ) ^ s * (Complex.Gamma s)⁻¹ * completed w s

abbreviation OAI.SevenEighths.HeckeFamily.K line 208
```
abbrev K := CyclotomicField 3 ℚ
```
abbreviation OAI.SevenEighths.HeckeFamily.O line 209
```
abbrev O := NumberField.RingOfIntegers K
```

definition OAI.SevenEighths.HeckeFamily.omega line 213

def omega : O := (IsCyclotomicExtension.zeta_spec 3 ℚ K).toInteger

definition OAI.SevenEighths.HeckeFamily.coordinateElement line 215

def coordinateElement (x y : ℤ) : O := (x : O) + (y : O) * omega

structure OAI.SevenEighths.HeckeFamily.Character line 217

structure Character where  modulus : Ideal O  modulus_ne_bot : modulus ≠ ⊥  residue : MulChar (O ⧸ modulus) ℂ  unit_trivial : ∀ u : Oˣ, residue (Ideal.Quotient.mk modulus (u : O)) = 1  period : ℕ  period_pos : 0 < period  period_mem : (period : O) ∈ modulus

definition OAI.SevenEighths.HeckeFamily.elementCoeff line 228

def elementCoeff (χ : Character) (z : O) : ℂ := χ.residue (Ideal.Quotient.mk χ.modulus z)

definition OAI.SevenEighths.HeckeFamily.coefficients line 230

def coefficients (χ : Character) (a : Fin χ.period × Fin χ.period) : ℂ :=  elementCoeff χ (coordinateElement a.1 a.2)

definition OAI.SevenEighths.HeckeFamily.continuedLattice line 233

def continuedLattice (χ : Character) : ℂ → ℂ := HeckeTheta.latticeL (coefficients χ)

definition OAI.SevenEighths.HeckeFamily.LFunction line 235

def LFunction (χ : Character) (s : ℂ) : ℂ := continuedLattice χ s / 6

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: 7/8, rational, hecke, zeta, re, character, zero, function and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sqrt and order.

Statements in The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$ come first because the AI draft reading of OpenAI’s note shares words with that paper’s title.

Theorem 1.1 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$

Theorem

Every finite-order Hecke L-function over F=\mathbb Q(\sqrt{-3}) has no zero in \Re s>7/8. The same holds for every Dirichlet L-function, including \zeta(s). A pole at s=1 for a principal character is allowed.

Section 1, Introduction Read it in the paper TeX source, line 106

Show the TeX

Every finite-order Hecke $L$-function over
$F=\mathbb Q(\sqrt{-3})$ has no zero in $\Re s>7/8$.
The same holds for every Dirichlet $L$-function, including $\zeta(s)$.
A pole at $s=1$ for a principal character is allowed.

Other possible paper matches, closest wording first

Proposition 5.1 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$. Words of the Lean statement, its names split into words, that also appear in the theorem: hx, kernel, quotient, complex, product, unit and 20 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounds and sqrt.
Proposition 2.1 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$. Words of the Lean statement, its names split into words, that also appear in the theorem: pi, continuation, numbers, omega, hecke, re and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: order.
Theorem 3.1 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$. Words of the Lean statement, its names split into words, that also appear in the theorem: rational, hecke, zeta, re, character, zero and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sqrt and order.
Proposition 20.3 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$. Words of the Lean statement, its names split into words, that also appear in the theorem: extension, 7/8, numbers, omega, one, zeta and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: order.
Proposition 11.3 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$. Words of the Lean statement, its names split into words, that also appear in the theorem: rational, hecke, re, one, zero, function and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sqrt and order.
Corollary 14.1 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$. Words of the Lean statement, its names split into words, that also appear in the theorem: ideals, using, completed, omega, coefficient, zeta and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sqrt.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 564 s; peak memory 6.6 GiB).
Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 003’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: No gap for zeta/Dirichlet L (Mathlib objects). Hecke 7/8 clause rests on a custom WeakFEPair-defined L-function (LFunction := continuedLattice χ s / 6) whose identification with the Dirichlet series is not in the statement; 11/12 companion not stated separately (implied).
Definitions the group asks readers to compare with the paper: Hecke part: def LFunction (χ : Character) (s : ℂ) : ℂ := continuedLattice χ s / 6 with continuedLattice χ = HeckeTheta.latticeL (coefficients χ) (custom theta/WeakFEPair construction; custom structure Character with modulus, residue : MulChar (O ⧸ modulus) ℂ, unit_trivial, period); no lemma linking it to Σ χ(a) N(a)^{-s}
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“It also establishes the same bound for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states part of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

245 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 62 to its own declarations, 181 to Mathlib’s documentation. 31 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAInoncomputable sectionopen Filter Asymptotics Set MeasureTheoryopen scoped Topology BigOperatorsnamespace SevenEighths.ThetaProducttheorem rapid_product_sub (f g : ℝ → ℂ) (a b : ℂ)    (hf : ∀ r : ℝ, (fun x => f x - a) =O[atTop] (fun x : ℝ => x ^ r))    (hg : ∀ r : ℝ, (fun x => g x - b) =O[atTop] (fun x : ℝ => x ^ r))    (r : ℝ) :    (fun x => f x * g x - a * b) =O[atTop] (fun x : ℝ => x ^ r) := by  have hg0 : g =O[atTop] (fun _ : ℝ => (1 : ℝ)) := by    have hc : (fun _ : ℝ => b) =O[atTop] (fun x : ℝ => x ^ (0 : ℝ)) := by      simpa using (isBigO_const_one ℝ b (atTop : Filter ℝ))    have h := (hg 0).add hc    simpa using h  have h₁ := (hf r).mul hg0  have h₂ := (hg r).const_mul_left a  have hid : (fun x => f x * g x - a * b) =      fun x => (f x - a) * g x + a * (g x - b) := by    funext x    ring  rw [hid]  have h₁' : (fun x => (f x - a) * g x) =O[atTop] (fun x : ℝ => x ^ r) := by    simpa using h₁  exact h₁'.add h₂def product (P Q : WeakFEPair ℂ)    (hPf : ContinuousOn P.f (Ioi 0)) (hPg : ContinuousOn P.g (Ioi 0))    (hQf : ContinuousOn Q.f (Ioi 0)) (hQg : ContinuousOn Q.g (Ioi 0)) :    WeakFEPair ℂ where  f := fun x => P.f x * Q.f x  g := fun x => P.g x * Q.g x  k := P.k + Q.k  ε := P.ε * Q.ε  f₀ := P.f₀ * Q.f₀  g₀ := P.g₀ * Q.g₀  hf_int := (hPf.mul hQf).locallyIntegrableOn measurableSet_Ioi  hg_int := (hPg.mul hQg).locallyIntegrableOn measurableSet_Ioi  hk := add_pos P.hk Q.hk  hε := mul_ne_zero P.hε Q.hε  h_feq x hx := by    rw [P.h_feq x hx, Q.h_feq x hx, Real.rpow_add hx]    simp only [smul_eq_mul, Complex.ofReal_mul]    ring  hf_top := rapid_product_sub P.f Q.f P.f₀ Q.f₀ P.hf_top Q.hf_top  hg_top := rapid_product_sub P.g Q.g P.g₀ Q.g₀ P.hg_top Q.hg_toptheorem rapid_rescale (f : ℝ → ℂ) (a : ℂ)    (hf : ∀ r : ℝ, (fun x => f x - a) =O[atTop] (fun x : ℝ => x ^ r))    {u : ℝ} (hu : 0 < u) (r : ℝ) :    (fun x => f (u * x) - a) =O[atTop] (fun x : ℝ => x ^ r) := by  have ht : Tendsto (fun x : ℝ => u * x) atTop atTop :=    (tendsto_const_mul_atTop_of_pos hu).2 tendsto_id  have h := (hf r).comp_tendsto ht  have heq : (fun x : ℝ => (u * x) ^ r) =ᶠ[atTop]      (fun x : ℝ => u ^ r * x ^ r) := by    filter_upwards [eventually_ge_atTop (0 : ℝ)] with x hx    exact Real.mul_rpow hu.le hx  exact h.trans (heq.isBigO.trans ((isBigO_refl (fun x : ℝ => x ^ r) atTop).const_mul_left _))def rescale (P : WeakFEPair ℂ)    (hPf : ContinuousOn P.f (Ioi 0)) (hPg : ContinuousOn P.g (Ioi 0))    (u : ℝ) (hu : 0 < u) : WeakFEPair ℂ where  f := fun x => P.f (u * x)  g := fun x => P.g (u⁻¹ * x)  k := P.k  ε := P.ε * (u ^ (-P.k) : ℝ)  f₀ := P.f₀  g₀ := P.g₀  hf_int := (hPf.comp (by fun_prop) (fun x hx => mul_pos hu hx)).locallyIntegrableOn    measurableSet_Ioi  hg_int := (hPg.comp (by fun_prop) (fun x hx => mul_pos (inv_pos.mpr hu) hx)).locallyIntegrableOn    measurableSet_Ioi  hk := P.hk  hε := mul_ne_zero P.hε (Complex.ofReal_ne_zero.mpr (Real.rpow_pos_of_pos hu _).ne')  h_feq x hx := by    have heq : u * (1 / x) = 1 / (u⁻¹ * x) := by field_simp    rw [heq, P.h_feq _ (mul_pos (inv_pos.mpr hu) hx),      Real.mul_rpow (inv_pos.mpr hu).le hx.le, Real.inv_rpow hu.le,      ← Real.rpow_neg hu.le]    simp only [smul_eq_mul, Complex.ofReal_mul]    ring  hf_top := rapid_rescale P.f P.f₀ P.hf_top hu  hg_top := rapid_rescale P.g P.g₀ P.hg_top (inv_pos.mpr hu)def shiftedPair (a b : UnitAddCircle) : WeakFEPair ℂ :=  product (HurwitzZeta.hurwitzEvenFEPair a) (HurwitzZeta.hurwitzEvenFEPair b)    (Complex.continuous_ofReal.comp_continuousOn (HurwitzZeta.continuousOn_evenKernel a))    (Complex.continuous_ofReal.comp_continuousOn (HurwitzZeta.continuousOn_cosKernel a))    (Complex.continuous_ofReal.comp_continuousOn (HurwitzZeta.continuousOn_evenKernel b))    (Complex.continuous_ofReal.comp_continuousOn (HurwitzZeta.continuousOn_cosKernel b))@[simp] theorem shiftedPair_k (a b : UnitAddCircle) : (shiftedPair a b).k = 1 := by  norm_num [shiftedPair, product, HurwitzZeta.hurwitzEvenFEPair]@[simp] theorem shiftedPair_f (a b : UnitAddCircle) (t : ℝ) :    (shiftedPair a b).f t =      (HurwitzZeta.evenKernel a t : ℂ) * (HurwitzZeta.evenKernel b t : ℂ) := rfltheorem shiftedPair_differentiableAt (a b : UnitAddCircle) {s : ℂ}    (hs₀ : s ≠ 0) (hs₁ : s ≠ 1) :    DifferentiableAt ℂ (shiftedPair a b).Λ s := by  exact (shiftedPair a b).differentiableAt_Λ (Or.inl hs₀)    (Or.inl (by simpa using hs₁))def scaledEvenPair (a : UnitAddCircle) (u : ℝ) (hu : 0 < u) : WeakFEPair ℂ :=  rescale (HurwitzZeta.hurwitzEvenFEPair a)    (Complex.continuous_ofReal.comp_continuousOn (HurwitzZeta.continuousOn_evenKernel a))    (Complex.continuous_ofReal.comp_continuousOn (HurwitzZeta.continuousOn_cosKernel a)) u hutheorem scaledEvenPair_f_continuous (a : UnitAddCircle) (u : ℝ) (hu : 0 < u) :    ContinuousOn (scaledEvenPair a u hu).f (Ioi 0) :=  (Complex.continuous_ofReal.comp_continuousOn    (HurwitzZeta.continuousOn_evenKernel a)).comp (by fun_prop)      (fun x hx => mul_pos hu hx)theorem scaledEvenPair_g_continuous (a : UnitAddCircle) (u : ℝ) (hu : 0 < u) :    ContinuousOn (scaledEvenPair a u hu).g (Ioi 0) :=  (Complex.continuous_ofReal.comp_continuousOn    (HurwitzZeta.continuousOn_cosKernel a)).comp (by fun_prop)      (fun x hx => mul_pos (inv_pos.mpr hu) hx)def rectangularPair (a b : UnitAddCircle) (u v : ℝ) (hu : 0 < u) (hv : 0 < v) :    WeakFEPair ℂ :=  product (scaledEvenPair a u hu) (scaledEvenPair b v hv)    (scaledEvenPair_f_continuous a u hu) (scaledEvenPair_g_continuous a u hu)    (scaledEvenPair_f_continuous b v hv) (scaledEvenPair_g_continuous b v hv)@[simp] theorem rectangularPair_k (a b : UnitAddCircle) (u v : ℝ)    (hu : 0 < u) (hv : 0 < v) : (rectangularPair a b u v hu hv).k = 1 := by  norm_num [rectangularPair, scaledEvenPair, rescale, product, HurwitzZeta.hurwitzEvenFEPair]end SevenEighths.ThetaProductnamespace SevenEighths.EisensteinThetaopen ThetaProductdef xShift (a b L : ℝ) (e : Fin 2) : ℝ := (2 * a - b - L * e) / (2 * L)def yShift (b L : ℝ) (e : Fin 2) : ℝ := (b + L * e) / (2 * L)def parityPair (a b L : ℝ) (hL : 0 < L) (e : Fin 2) : WeakFEPair ℂ :=  rectangularPair (xShift a b L e) (yShift b L e) (L ^ 2) (3 * L ^ 2)    (sq_pos_of_pos hL) (mul_pos (by norm_num) (sq_pos_of_pos hL))end SevenEighths.EisensteinThetanamespace SevenEighths.HeckeThetaopen EisensteinThetatheorem locallyIntegrableOn_sum {ι : Type*} (s : Finset ι)    (f : ι → ℝ → ℂ) (hf : ∀ i ∈ s, LocallyIntegrableOn (f i) (Ioi 0)) :    LocallyIntegrableOn (∑ i ∈ s, f i) (Ioi 0) := by  exact Finset.sum_induction f (fun g => LocallyIntegrableOn g (Ioi 0))    (fun _ _ => LocallyIntegrableOn.add) (locallyIntegrableOn_const 0) hfdef finitePair {ι : Type*} [Fintype ι] (P : ι → WeakFEPair ℂ)    (hk : ∀ i, (P i).k = 1) (w : ι → ℂ) : WeakFEPair ℂ where  f := ∑ i, w i • (P i).f  g := ∑ i, (w i * (P i).ε) • (P i).g  k := 1  ε := 1  f₀ := ∑ i, w i * (P i).f₀  g₀ := ∑ i, w i * (P i).ε * (P i).g₀  hf_int := locallyIntegrableOn_sum Finset.univ _    (fun i _ => (P i).hf_int.smul (w i))  hg_int := locallyIntegrableOn_sum Finset.univ _    (fun i _ => (P i).hg_int.smul (w i * (P i).ε))  hk := by norm_num  hε := one_ne_zero  h_feq x hx := by    simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, one_mul]    rw [Finset.mul_sum]    apply Finset.sum_congr rfl    intro i _    rw [(P i).h_feq x hx, hk i]    simp only [smul_eq_mul]    ring  hf_top r := by    have h := Asymptotics.IsBigO.fun_sum (s := Finset.univ)      (fun i _ => ((P i).hf_top r).const_mul_left (w i))    simpa only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, ← Finset.sum_sub_distrib,      mul_sub] using h  hg_top r := by    have h := Asymptotics.IsBigO.fun_sum (s := Finset.univ)      (fun i _ => ((P i).hg_top r).const_mul_left (w i * (P i).ε))    simpa only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, ← Finset.sum_sub_distrib,      mul_sub] using hdef pair {N : ℕ} [NeZero N] (w : Fin N × Fin N → ℂ) : WeakFEPair ℂ :=  finitePair (fun p : (Fin N × Fin N) × Fin 2 =>    parityPair p.1.1 p.1.2 N (Nat.cast_pos.mpr (NeZero.pos N)) p.2)    (fun _ => ThetaProduct.rectangularPair_k ..) (fun p => w p.1)def completed {N : ℕ} [NeZero N] (w : Fin N × Fin N → ℂ) : ℂ → ℂ := (pair w).Λdef latticeL {N : ℕ} [NeZero N] (w : Fin N × Fin N → ℂ) (s : ℂ) : ℂ :=  (Real.pi : ℂ) ^ s * (Complex.Gamma s)⁻¹ * completed w send SevenEighths.HeckeThetanamespace SevenEighths.HeckeFamilyabbrev K := CyclotomicField 3 ℚabbrev O := NumberField.RingOfIntegers Kinstance : IsCyclotomicExtension {3} ℚ K := CyclotomicField.isCyclotomicExtension 3 ℚdef omega : O := (IsCyclotomicExtension.zeta_spec 3 ℚ K).toIntegerdef coordinateElement (x y : ℤ) : O := (x : O) + (y : O) * omegastructure Character where  modulus : Ideal O  modulus_ne_bot : modulus ≠ ⊥  residue : MulChar (O ⧸ modulus) ℂ  unit_trivial : ∀ u : Oˣ, residue (Ideal.Quotient.mk modulus (u : O)) = 1  period : ℕ  period_pos : 0 < period  period_mem : (period : O) ∈ modulusinstance (χ : Character) : NeZero χ.period := ⟨Nat.ne_of_gt χ.period_pos⟩def elementCoeff (χ : Character) (z : O) : ℂ := χ.residue (Ideal.Quotient.mk χ.modulus z)def coefficients (χ : Character) (a : Fin χ.period × Fin χ.period) : ℂ :=  elementCoeff χ (coordinateElement a.1 a.2)def continuedLattice (χ : Character) : ℂ → ℂ := HeckeTheta.latticeL (coefficients χ)def LFunction (χ : Character) (s : ℂ) : ℂ := continuedLattice χ s / 6theorem LFunction_ne_zero_of_seven_eighths_lt_re    (χ : Character) {s : ℂ} (hs : (7 / 8 : ℝ) < s.re)    (hpole : ¬ (χ.residue = 1 ∧ s = 1)) : LFunction χ s ≠ 0 := by sorryend SevenEighths.HeckeFamilyendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$

Theorem

Section 1, Introduction Read it in the paper TeX source, line 106

Show the TeX

Every finite-order Hecke $L$-function over
$F=\mathbb Q(\sqrt{-3})$ has no zero in $\Re s>7/8$.
The same holds for every Dirichlet $L$-function, including $\zeta(s)$.
A pole at $s=1$ for a principal character is allowed.

Corollary

There are absolute constants A,C>0 such that, for every odd prime p, the least positive quadratic nonresidue n(p) satisfies

n(p)\le C(\log p)^A.

In particular, n(p)\ll_\delta p^\delta for every \delta>0, proving Vinogradov’s least quadratic nonresidue conjecture [TaoNonresidues, Conjecture 1.1]. Given an odd prime p and a\in\mathbb F_p in binary representation, there is a deterministic algorithm, with running time polynomial in \log p, that returns a square root of a or reports that none exists.

Section 1, Introduction Read it in the paper TeX source, line 325

Show the TeX

There are absolute constants $A,C>0$ such that, for every odd prime $p$,
the least positive quadratic nonresidue $n(p)$ satisfies
\[
 n(p)\le C(\log p)^A.
\]
In particular, $n(p)\ll_\delta p^\delta$ for every $\delta>0$, proving
Vinogradov's least quadratic nonresidue conjecture
\cite[Conjecture 1.1]{TaoNonresidues}.
Given an odd prime $p$ and $a\in\mathbb F_p$ in binary representation,
there is a deterministic algorithm, with running time polynomial in
$\log p$, that returns a square root of $a$ or reports that none exists.

Uniform exclusion of Landau–Siegel zeros

Theorem

There is an absolute constant c>0 such that every real zero \beta\in(0,1) of every primitive nonprincipal real Dirichlet L-function of conductor q\ge3 satisfies

(1-\beta)\log q\ge c.

Section 1, Introduction Read it in the paper TeX source, line 66

Show the TeX

There is an absolute constant $c>0$ such that every real zero
$\beta\in(0,1)$ of every primitive nonprincipal real Dirichlet
$L$-function of conductor $q\ge3$ satisfies
\begin{equation}
 (1-\beta)\log q\ge c.
\end{equation}

Configuration and file facts

File: lean/ComparatorChallenges/HeckeSevenEighths.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/NumberTheory/DirichletL/Hecke/Nonvanishing.lean
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 23 local vocabulary declarations
Lines of Lean, without comments or blank lines: 185
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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