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The Coprimality Density Framework

The Φ(N) Convergence to ζ(2)−1

Φ(N) = (1/N) ∑m=1N φ(m)/m  →  6/π²  (proved)    Eζ(N) = Φ(N) − 6/π² = O(log N / N) (conjecture)

Each chart is computed live. For each integer M, the ratio φ(M)/M measures what fraction of residues are coprime to M. Primes spike near 1; highly composite numbers sink toward 0. Their average converges to 6/π² = 1/ζ(2) — and the error term Eζ(N) reveals how fast, driven entirely by the primes that pull the sum above the limit.

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Convergence to 6/π²
Chart 01 Φ(N) converging to 6/π² — cumulative coprimality density Running average of φ(m)/m. Primes (φ(p)/p = (p−1)/p near 1) spike upward; smooth numbers sink. The gold dashed line is the target 6/π² ≈ 0.60793. The convergence is proved; the rate is not.
Φ(N)
6/π² target
Prime positions
Chart 02 φ(M)/M per integer — primes vs composites The raw φ(M)/M for each M. Primes (gold) cluster near 1 because φ(p) = p−1. Highly composite numbers (teal, low ratio) pull the average down. The horizontal line at 6/π² shows where the average ultimately lands. The isolation effect: remove all primes and the remaining average falls strictly below 6/π².
φ(p)/p (prime)
φ(M)/M (composite)
6/π² line
Error term Eζ(N) = Φ(N) − 6/π²
Chart 03 Eζ(N) — signed error with prime markers Signed deviation Φ(N) − 6/π². Gold ticks mark primes — each prime causes an upward spike in Φ(N), briefly pushing Eζ positive. Between primes, smooth numbers pull it back down. Sign changes (coral dots) mark the oscillation crossings of zero.
Eζ(N)
Prime tick
Sign change
Chart 04 Eζ(N) · N — testing O(1/N) conjecture If Eζ(N) = O(1/N) then Eζ(N)·N should be bounded. Under RH the known conditional bound is O(N−1/2+ɛ), making Eζ(N)·N grow slowly. The empirical behaviour here tells you whether O(1/N) holds or whether the actual rate is slower.
Eζ(N)·N
log N ref
Chart 05 log|Eζ(N)| vs log N — convergence rate diagnostic Log-log plot of the error magnitude. The slope gives the empirical convergence rate. Slope = −1 means O(1/N). Slope = −1/2 means O(1/√N) (RH conditional bound). The OLS regression line shows where the data actually sits.
log|Eζ|
Slope = −1 ref (O(1/N))
Slope = −1/2 ref (RH bound)
OLS fit
Prime isolation & C comparison
Chart 06 Prime isolation — Φ(N) with and without primes Two running averages: one including all integers (gold, converges to 6/π²), one excluding primes (coral, converges to a strictly lower value). The gap between them is the prime isolation effect — the primes are exactly what pulls the ratio above the composite-only limit and into 6/π².
Φ(N) all integers → 6/π²
Φ(N) composites only
6/π² target
Chart 07 Eζ(N) vs E(N) — two error terms, one framework Overlay of the ζ(2) error Eζ(N) = Φ(N) − 6/π² (gold) and the lift survival error E(N) = R(N) − C (teal) from the C framework. Both are driven by primes; both oscillate around zero; both connect to ζ(s). Their relative oscillation structure is an open question.
Eζ(N) = Φ(N) − 6/π²
E(N) = R(N) − C (scaled ×100)
Analytic results & open questions
Theorem — Convergence (classical)
(1/N) ∑m=1N φ(m)/m  →  6/π² = 1/ζ(2) = ∏p(1 − 1/p²)    ■
Proof sketch. By Euler’s product, ∑m≤N φ(m)/m = ∑m≤N ∏p|m(1−1/p). By Möbius inversion, φ(m)/m = ∑d|m μ(d)/d. Summing over m≤N and swapping order gives (1/N)∑φ(m)/m → ∑d μ(d)/d² = 1/ζ(2) = 6/π². ■
Conjecture — Rate (open)
Eζ(N) = (1/N) ∑m=1N φ(m)/m  −  6/π²  =  O(log N / N)    (numerical conjecture)
What is known. The error Eζ(N) is related to the Mertens function M(N) = ∑m≤N μ(m). Under the Riemann Hypothesis, M(N) = O(N1/2+ɛ) for all ɛ > 0, which implies Eζ(N) = O(N−1/2+ɛ). Unconditionally, Eζ(N) = O(exp(−c√log N)) by the prime number theorem. The empirical rate seen in this explorer — consistent with O(log N / N) — would imply M(N) = O(log N) unconditionally, which is far stronger than anything proved. This is noted as a numerical observation, not a claim.

Connection to C framework. The constant C = ∏p(p²−2)/(p²−1) = ζ(2)·dFT and 6/π² = 1/ζ(2) = ∏p(1−1/p²) are both Euler products over the primes, both proved, both approached from above by the respective ratios. The error terms E(N) and Eζ(N) are structurally parallel — driven by the same arithmetic, oscillating around the same primes, connected to the same L-functions.
The Prime Isolation Effect
Let Φcomp(N) = (1/|{m≤N: m composite}|) ∑m≤N, m composite φ(m)/m. Numerically, Φcomp(N) converges to a value strictly below 6/π². The primes — with φ(p)/p = (p−1)/p → 1 — are exactly the integers that pull the average up to 6/π². Removing them lowers the limit. This is the isolation property: the primes are the unique class of integers whose coprimality density approaches 1, and their density in the integers (by PNT, 1/log N) is precisely sufficient to raise the composite average to 1/ζ(2). Chart 06 makes this visible.