primeLedgerPartition_formal

formal source

  53  ∃ F : (Finset ℕ → ℂ), ∀ S, F S = finitePrimeLedgerPartition s S
  54
  55/-- The formal partition exists, by finite partial products. -/
  56theorem primeLedgerPartition_formal (s : ℂ) : PrimeLedgerPartition s :=
  57  ⟨fun S => finitePrimeLedgerPartition s S, fun _ => rfl⟩
  58
  59/-- Analytic certificate connecting the formal prime-ledger partition to zeta
  60on a convergence domain. -/
  61structure EulerLedgerPartitionCert where
  62  /-- Euler product agrees with `riemannZeta` for `Re(s) > 1`, expressed as
  63  convergence of finite prime-ledger partitions. -/
  64  eulerProduct_eq_zeta :
  65    ∀ s : ℂ, 1 < s.re →
  66      Filter.Tendsto (fun n : ℕ ↦ finitePrimeLedgerPartition s (Nat.primesBelow n))
  67        Filter.atTop (𝓝 (riemannZeta s))
  68  /-- The formal prime-ledger partition exists at every complex scale. -/
  69  formal_partition : ∀ s : ℂ, PrimeLedgerPartition s
  70  /-- Primes are exactly the ledger atoms. -/
  71  prime_atoms : PrimeLedgerCert
  72
  73/-- The structural Euler ledger certificate.  The analytic equality field now
  74uses Mathlib's Euler-product theorem for `riemannZeta` on `Re(s) > 1`; the
  75finite products are exactly the finite prime-ledger partitions. -/
  76def eulerLedgerPartitionCert : EulerLedgerPartitionCert where
  77  eulerProduct_eq_zeta := by
  78    intro s hs
  79    have hmathlib :
  80        Filter.Tendsto (fun n : ℕ ↦ ∏ p ∈ Nat.primesBelow n, (1 - (p : ℂ) ^ (-s))⁻¹)
  81          Filter.atTop (𝓝 (riemannZeta s)) :=
  82      _root_.riemannZeta_eulerProduct hs
  83    have hpart : (fun n : ℕ ↦ finitePrimeLedgerPartition s (Nat.primesBelow n)) =
  84        (fun n : ℕ ↦ ∏ p ∈ Nat.primesBelow n, (1 - (p : ℂ) ^ (-s))⁻¹) := by
  85      funext n
  86      unfold finitePrimeLedgerPartition primeLedgerLocalPartition primePostingWeight