IndisputableMonolith.NumberTheory.EulerProductEqualsZeta
This module restates the classical Euler product for the Riemann zeta function inside the Recognition Science prime-ledger namespace. It records that finite products over primes converge to zeta(s) for Re(s) > 1, linking the ledger partitions of EulerLedgerPartition with the logic-native primes of LogicPrimeLedgerAtom. The structure isolates the infinite-product step as an analytic boundary rather than an RS derivation. Number theorists working on the Riemann hypothesis within the RS recovered-number stack would cite it to import the standard z
claimFinite prime partial products converge to the Riemann zeta function: for Re(s) > 1, lim_{N→∞} ∏_{p≤N} (1−p^{−s})^{−1} = ζ(s), where the product runs over primes in the RS prime ledger.
background
The module sits in the NumberTheory domain and treats the Euler product as the partition function of independent prime-ledger postings. It imports EulerLedgerPartition, whose doc-comment states that finite product statements are proved directly while the infinite equality with riemannZeta is isolated in EulerLedgerPartitionCert because the exact Mathlib API is an analytic import boundary. It also imports LogicPrimeLedgerAtom, which supplies logic-native prime ledger atoms and transports primality through the recovery equivalence LogicNat.toNat. The local convention is to keep Mathlib ℂ as the analytic substrate.
proof idea
This is a module that imports the partition and atom layers, then restates the Euler-product theorems under the RS namespace. Finite cases are handled directly by the upstream EulerLedgerPartition; the infinite equality is isolated as an analytic boundary condition rather than re-proved inside RS.
why it matters in Recognition Science
The module supplies the Euler-product equality required by the downstream LogicComplexCompat compatibility layer. That layer explicitly uses Mathlib ℂ for holomorphy and contour integration while embedding the zeta function in the recovered-number stack; the present module therefore closes the analytic-number-theory link needed for Riemann-hypothesis work on the prime-ledger side.
scope and limits
- Does not derive the Euler product from RS axioms alone.
- Does not address convergence for Re(s) ≤ 1.
- Does not redefine the zeta function or its analytic properties.
- Does not extend the ledger partition beyond prime factors.