finitePrimeLedgerPartition_insert_nonprime

formal source

  34  ∏ p ∈ S, if Nat.Prime p then primeLedgerLocalPartition s p else 1
  35
  36/-- Finite prime-ledger partition is insensitive to non-prime entries. -/
  37theorem finitePrimeLedgerPartition_insert_nonprime
  38    (s : ℂ) (S : Finset ℕ) {n : ℕ} (hnS : n ∉ S) (hn : ¬ Nat.Prime n) :
  39    finitePrimeLedgerPartition s (insert n S) = finitePrimeLedgerPartition s S := by
  40  unfold finitePrimeLedgerPartition
  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))