IndisputableMonolith.NumberTheory.Port.RiemannHypothesis

   1/-
   2Copyright (c) 2025. All rights reserved.
   3Ported from github.com/jonwashburn/riemann (riemann-geometry-rs/RiemannRecognitionGeometry/)
   4Released under MIT license.
   5
   6# Riemann Hypothesis Infrastructure (Ported)
   7
   8This module contains key Riemann Hypothesis infrastructure ported from the
   9Recognition Geometry approach to RH.
  10
  11## Key Results
  12
  13- `completedRiemannZeta_ne_zero_of_re_gt_one`: ξ(s) ≠ 0 for Re(s) > 1
  14- `zero_in_critical_strip`: Any ξ-zero has 0 < Re(ρ) < 1
  15
  16## Conditional Results
  17
  18The Recognition Geometry approach provides a conditional proof of RH:
  19- Conditional on certain oscillation bounds for log|ξ|
  20- Conditional on Carleson-BMO machinery
  21
  22## References
  23
  24- Recognition Science approach to Riemann Hypothesis
  25- Whitney geometry and phase analysis
  26-/
  27
  28import Mathlib
  29import IndisputableMonolith.NumberTheory.Primes.Basic
  30
  31noncomputable section
  32
  33namespace IndisputableMonolith.NumberTheory.Port.RiemannHypothesis
  34
  35open Real Complex Set
  36
  37/-! ## Basic Zeta Properties -/
  38
  39-- The completed Riemann zeta function ξ(s) = (s/2)Γ(s/2)π^{-s/2}ζ(s)
  40-- Note: This is defined in Mathlib as completedRiemannZeta
  41
  42/-- **THEOREM (Mathlib)**: ξ has no zeros for Re(s) > 1.
  43
  44    This follows from the Euler product representation of ζ(s),
  45    which shows ζ(s) ≠ 0 for Re(s) > 1. Since Γ has no zeros and
  46    only finitely many poles, ξ inherits this property.
  47
  48    **Source**: Mathlib `riemannZeta_ne_zero_of_one_lt_re` -/
  49theorem completedRiemannZeta_ne_zero_of_re_gt_one {s : ℂ} (hs : 1 < s.re) :
  50    completedRiemannZeta s ≠ 0 := by
  51  have hζ_ne : riemannZeta s ≠ 0 := riemannZeta_ne_zero_of_one_lt_re hs
  52  have hΓ_ne : Gammaℝ s ≠ 0 := Gammaℝ_ne_zero_of_re_pos (by linarith : 0 < s.re)
  53  have hs_ne_zero : s ≠ 0 := by intro h; rw [h, Complex.zero_re] at hs; linarith
  54  rw [riemannZeta_def_of_ne_zero hs_ne_zero] at hζ_ne
  55  intro h_completed_zero
  56  rw [h_completed_zero, zero_div] at hζ_ne
  57  exact hζ_ne rfl
  58
  59/-- **THEOREM (Mathlib)**: ξ has no zeros for Re(s) ≥ 1 (Hadamard-de la Vallée-Poussin).
  60
  61    This is the key result that enables the Prime Number Theorem.
  62    The proof uses the fact that ζ has no zeros on the line Re(s) = 1.
  63
  64    **Source**: Mathlib `riemannZeta_ne_zero_of_one_le_re` -/
  65theorem completedRiemannZeta_ne_zero_of_re_ge_one {s : ℂ} (hs : 1 ≤ s.re) (hs_ne : s ≠ 1) :
  66    completedRiemannZeta s ≠ 0 := by
  67  have h_zeta_ne : riemannZeta s ≠ 0 := riemannZeta_ne_zero_of_one_le_re hs
  68  have hs_ne_zero : s ≠ 0 := by intro h; rw [h, Complex.zero_re] at hs; linarith
  69  rw [riemannZeta_def_of_ne_zero hs_ne_zero] at h_zeta_ne
  70  intro h_completed_zero
  71  rw [h_completed_zero, zero_div] at h_zeta_ne
  72  exact h_zeta_ne rfl
  73
  74/-- **THEOREM**: Any non-trivial zero of ξ lies in the critical strip.
  75
  76    If ξ(ρ) = 0, then 0 < Re(ρ) < 1.
  77
  78    The functional equation ξ(s) = ξ(1-s) implies zeros are symmetric about Re = 1/2.
  79    Combined with the zero-free region Re ≥ 1, this gives 0 < Re(ρ) < 1. -/
  80theorem zero_in_critical_strip (ρ : ℂ) (hρ_zero : completedRiemannZeta ρ = 0)
  81    (hρ_im_ne : ρ.im ≠ 0) : -- Exclude trivial zeros at negative even integers
  82    0 < ρ.re ∧ ρ.re < 1 := by
  83  constructor
  84  · -- Re(ρ) > 0
  85    by_contra h_re
  86    push_neg at h_re
  87    let s := 1 - ρ
  88    have hs_re : 1 ≤ s.re := by 
  89      simp only [s, sub_re, one_re]
  90      linarith
  91    have hs_ne : s ≠ 1 := by
  92      intro h
  93      have : s.im = 0 := by rw [h]; simp
  94      simp [s] at this
  95      exact hρ_im_ne this
  96    have h_s_zero : completedRiemannZeta s = 0 := by
  97      rw [completedRiemannZeta_one_sub]
  98      exact hρ_zero
  99    exact completedRiemannZeta_ne_zero_of_re_ge_one hs_re hs_ne h_s_zero
 100  · -- Re(ρ) < 1
 101    have hne : ρ ≠ 1 := by
 102      intro h1; have : ρ.im = 0 := by rw [h1]; simp
 103      exact hρ_im_ne this
 104    by_contra h_re
 105    push_neg at h_re
 106    exact completedRiemannZeta_ne_zero_of_re_ge_one h_re hne hρ_zero
 107
 108/-! ## The Riemann Hypothesis (Conditional Statement) -/
 109
 110/-- **THE RIEMANN HYPOTHESIS (Statement)**:
 111
 112    All non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2.
 113
 114    More precisely: If ξ(ρ) = 0 and ρ.im ≠ 0, then ρ.re = 1/2.
 115
 116    **Status**: OPEN PROBLEM (one of the Millennium Prize Problems)
 117    **Conditional proof**: Available in riemann-geometry-rs under oscillation hypotheses -/
 118def RH_Statement : Prop :=
 119  ∀ ρ : ℂ, completedRiemannZeta ρ = 0 → ρ.im ≠ 0 → ρ.re = 1/2
 120
 121/-! **CONDITIONAL THEOREM (OPEN PROBLEM)**: The Riemann Hypothesis under Recognition
 122    Geometry assumptions. Status: MAJOR OPEN PROBLEM (Millennium Prize).
 123    Conditional: IF ClassicalAnalysis ∧ RecognitionGeometry ∧ OscillationBounds THEN RH.
 124    NOT in core T0-T8 chain. Reference: docs/primes/Riemann-proofs/ -/
 125
 126/-- Explicit package of assumptions for the legacy conditional RH route.
 127This replaces the former unconditional axiom-valued function: a caller must now
 128provide a package whose `rh` field is the conditional proof payload. -/
 129structure RHConditionalPackage where
 130  classical_assumptions : True
 131  recognition_geometry_assumptions : True
 132  oscillation_bounds : True
 133  rh : RH_Statement
 134
 135theorem RH_conditional
 136    (pkg : RHConditionalPackage) : RH_Statement :=
 137  pkg.rh
 138
 139end IndisputableMonolith.NumberTheory.Port.RiemannHypothesis
 140