Two structures of the prime numbers are examined in parallel: a geometric diagram of modular residue arithmetic organized by the Farey sequence, and a factorization of the Euler product ζ(2) = π²/6 by prime gap class. Neither constitutes a proof of any conjecture. Both are verified computations on classical objects.
The diagram places every rational r/m with 1 ≤ m ≤ N at angle 2πr/m on ring m, making the Farey sequence geometric. Fractions with gcd(r,m) = 1 (blue) are the reduced Farey fractions. By the Franel-Landau theorem, any systematic bias in their angular distribution — measured by the discrepancy DN — implies a zero of ζ(s) off the critical line. Their observed near-uniformity at finite N is consistent with RH, not evidence for it in a formal sense. The composite residues gcd(r,m) > 1 (red) concentrate at angles that are rational multiples of 2π, tracing the multiplicative structure of the integers across rings.
Cross-mod connections follow each residue r through rings N→1. With gap g active, the overlay marks the pairs (p mod N, (p+g) mod N) for primes with forward gap exactly g. Outer-ring chords show which residue classes host twin, cousin, or sexy prime pairs; the inward spirals show the full divisibility structure beneath each endpoint.
Euler's product ζ(2) = π²/6 = ∏p p²/(p²−1) converges absolutely, so its factors may be rearranged freely. Grouping by forward gap — the distance from each prime to the next — gives ζ(2) = ∏g Pg, where Pg = ∏{p : gap(p)=g} p²/(p²−1). This is a reorganization of a known result, not a theorem. Its value is numerical: it makes the weight of each gap class in the Euler product directly measurable.
At N = 400 million, gap class 1 (the prime 2 alone, contributing the fixed factor 4/3) accounts for 57.8% of log ζ(2). Gap class 2 (twin primes) accounts for a further 34.8%. Together they explain over 92% of the total. A structural question follows: if the twin prime set were finite, P2(s) would be a finite product of factors ps/(ps−1), holomorphic on all of ℂ and bounded away from zero for Re(s) > 0. The remaining sub-products would then have to account for the poles and zeros of ζ(s) entirely — whether this is analytically possible via the identity ζ(s) = ∏g Pg(s) appears to be open.
G = {1, 2, 4, 6, 8, …} is the set of all realized prime gap sizes. Pg = ∏{p : gap(p)=g} p²/(p²−1).
Define gap(pn) = pn+1 − pn. By absolute convergence:
Gap class g = 1 contains only the prime 2, since gap(2) = 3 − 2 = 1, giving the factor 4/3 exactly. Every other realized gap is even, since all primes beyond 2 are odd.
The gap-class factorization is a rearrangement of Euler's product, valid by absolute convergence. It is not a new theorem and proves nothing about the distribution of primes. The finiteness question is a structural observation — open, and not a proof of the twin prime conjecture.
Quick configurations for common research scenarios
To use pre-computed prime lists — note the same ~300M browser limit applies to uploaded files
The non-trivial zeros ρ = ½ + itⁿ of ζ(s) govern oscillations in the prime-counting function and, through the Franel–Landau theorem, in the Farey discrepancy. This section has three linked views — all controlled by the same Mod N and Zeros inputs above.
● Circle canvas — Blue dots: gcd(r,m)=1 Farey points at angle 2πr/m on ring m. The purple dashed line is the critical axis Re(s)=½. Pink ticks on that axis mark each zero height tⁿ at radial depth r = tⁿ/N·rmax — so tⁿ = N sits exactly on the outer ring. As you increase Mod N, earlier zeros sink inward and new ones appear. This is a radial depth encoding, not a phase encoding.
● DFT Spectrum — Discrete Fourier Transform of the gcd=1 angular density at mod N. The horizontal axis is DFT frequency k; gold dashed lines mark k = tⁿ/(2π) for each zero. The empirical observation is that spectrum weight concentrates near these frequencies as N grows — a consequence of the explicit formula, which writes the prime-counting error as a sum over zeros each contributing oscillation at rate tⁿ.
● Discrepancy vs RH bound — Purple curve: actual Farey discrepancy DN = maxx|#{r/m ∈ FN : r/m ≤ x}/|FN| − x|, computed for N = 2…Mod N. Gold dashed: the RH asymptotic envelope log(N)/(2π√N). Grey dashed: trivial bound 3/(N+2). Under RH, DN is eventually O(N−½ log N) — the gold line is that leading term. Pink vertical lines mark N = ⌊tⁿ⌋; the discrepancy visibly oscillates near these heights. The main canvas ζ button shows a different but related view: phase points tⁿ·log(N) mod 2π on the outer ring, where the angular clustering of those phases encodes the Franel sum discrepancy direction.
Generate a comprehensive image combining selected charts with analysis summary and mathematical narrative
These settings apply to ALL chart exports (individual and composite)
The gap-class decomposition above reorganises the Euler product ζ(2) = ∏ p²/(p²−1) by prime gap families. The Mertens function below probes the same prime landscape from a different angle: the Möbius function μ(n) encodes the exact prime-factorisation structure that drives those products, with μ(n) = ±1 for square-free n and 0 otherwise.
Their deep link is the Riemann Hypothesis. The RH is equivalent to both:
Both are measuring the same thing: whether prime factorizations carry systematic bias. The convergence in the gap products and the ±√n bound on M(n) are two formulations of the same conjecture.
| n | μ(n) | M(n) | ΔM | Square-Free | Prime Factors | |M(n)| | Within ±√n | Status |
|---|---|---|---|---|---|---|---|---|
| Σμ(n) = — | Final: — | Sq-Free: — | Avg |M|: — | Within bound: — | Zeros: — | |||
This work builds on Euler's 1737 product formula for ζ(2) = π²/6 (Basel problem, first solved by Euler in 1734). The gap-class factorization ζ(2) = ∏g Pg follows directly from absolute convergence of the Euler product — it is a reorganization of a known result, not a new theorem. Its value is computational and structural.
The question of whether a finite twin prime set would be analytically incompatible with ζ(s)'s structure via this product identity is raised as an open structural question — not a proof of the Twin Prime Conjecture.
Established (prior literature): Euler product for ζ(s) (Euler, 1737); Basel problem solution π²/6 (Euler, 1734); twin prime constant C2 ≈ 0.66016; Cramér's model for prime gaps (1936).
This work: The specific partition of the Euler product by forward gap class and its numerical analysis; the structural question about analytic obstructions to a finite twin prime set via the product identity; computational data at N up to 300M (browser) with verified estimates to 400M.
The author makes no claim that these observations constitute a proof of any unsolved conjecture. They are presented as computational evidence and structural observations for further investigation.
Provided for educational and research purposes. Use of this tool or citation of its findings in academic work should attribute Wessen Getachew and reference the gap-class decomposition framework.