finitePrimeLedgerPartition_insert_prime

formal source

  41  simp [hnS, hn]
  42
  43/-- Inserting a new prime multiplies the partition by its local factor. -/
  44theorem finitePrimeLedgerPartition_insert_prime
  45    (s : ℂ) (S : Finset ℕ) {p : ℕ} (hpS : p ∉ S) (hp : Nat.Prime p) :
  46    finitePrimeLedgerPartition s (insert p S) =
  47      primeLedgerLocalPartition s p * finitePrimeLedgerPartition s S := by
  48  unfold finitePrimeLedgerPartition
  49  simp [hpS, hp]
  50
  51/-- The Euler ledger partition as a formal infinite product over prime atoms. -/
  52def PrimeLedgerPartition (s : ℂ) : Prop :=
  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