Euler Product — Discrete Rational Structure

Status: active Domain: math Source: → wiki/research/rational-makeover-survey/ (§1)

What We Know

The first 3 primes {2, 3, 5} capture 95%+ of ζ(s) at integer s — structure is massively concentrated in the smallest primes

s	N=3 primes	N=10	N=50
2 (integer)	95.0%	99.3%	99.95%
3/2 (prime ratio)	81.1%	94.0%	98.9%
7/5 (prime ratio)	75.1%	90.6%	98.0%

At rational s = p/q near 1, convergence is much slower — more primes contribute meaningfully. Near the critical strip, prime structure is maximally distributed.
Prime-indexed truncation points (N = 2, 3, 5, 7…) show larger jumps in the partial product than composite-indexed points. The Euler product “breathes” at prime truncations.
At s = 7/5 (closest to critical line), the first prime alone captures only 52.8% — remaining 47% distributed across all other primes
Analytic continuation to complex s “smears” discrete prime-breathing into a smooth function. Restricting to rational s reveals discrete prime structure most clearly.
The “1-2% irrationality penalty” from Q-ZZ-01 may be recoverable by working at rational zeta arguments

Q-EM-01: Do partial Euler products at s = p/q predict v3/v4 resonance better than full ζ values?
Q-EM-02: Is there a “discrete rational zeta” function defined only at s ∈ ℚ with cleaner prime structure than ζ(s)?
Q-EM-03: At s = 3/2, do partial products at N = {2,3,5} form a coprime triple?
What happens at s = 1 + 1/p for various primes p? (Approach to the pole along prime rationals)

[[zeta-zeros-physical]] — extends (strong): Rational s values preserve arithmetic structure that complex continuation destroys; connects to the “lossy encoding” insight
[[coprimality]] — supports (moderate): The 95% concentration in {2,3,5} means coprimality of the first three primes governs most of ζ(s)
[[prime-harmonic-transform]] — depends-on (strong): PHT weight function IS the Euler product; PHT with w(n)=1/nˢ is literally a Dirichlet series evaluation
[[square-root-problem]] — extends (moderate): Rational s values reveal structure that irrational/complex s hides
[[six-dimensional-scaffold]] — supports (moderate): {2,3,5} = first three primes; 2×3=6 is the scaffold, and they capture 95% of ζ

If combined with [[prime-harmonic-transform]], the Euler product structure could inform better PHT weight functions
A “discrete rational zeta” could provide a cleaner mathematical foundation for prime resonance theory
The 95% concentration result is pedagogically powerful — “three primes rule the zeta function”