Each chart is computed live from the Euler-product definition of C. Move the slider and press Compute to see how E(N) = R(N) − C behaves. The eight charts probe convergence rate, oscillation structure, prime correlations, and the χ⊂4; restriction in turn.
R(N) → C (proved) E(N) = O(1/N) (conjectured)
Compute up to N
500
MAX
zooms in at 75% convergence
Computed N
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Constant C
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c₁ estimate
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Sign changes
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π(N) primes
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SC / π(N)
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Prime-at-SC
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Longest SC gap
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log|E| slope (OLS)
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Convergence — three views
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Set N and click Compute → to render all eight charts.
Chart 01|E(N)| with A/log N envelopeAbsolute error with fitted envelope. Consistent with O(1/N) conjecture — faster decay than 1/log N seen empirically. Rate unproved analytically.
Chart 02E(N) · N → bounded oscillationE(N)·N is bounded and oscillating — verified to N=2,000,000. Consistent with O(1/N) conjecture. The c₁/log N heuristic is shown for reference; comparison does not prove the rate.
Chart 03log|E(N)| vs log(log N) · slope → −1Log-log diagnostic. Empirical slope steeper than −1, consistent with O(1/N) conjecture. Sign-change breaks visible. Slope alone does not establish the rate analytically.
Sign change analysis
Chart 04Signed E(N) · prime markers · sign-change dotsFull signed oscillation of E(N). Gold ticks on the baseline mark every prime. Coral dots mark sign changes. Most sign changes fall near primes — but not every prime triggers one. At large N, gold ticks are thinned to every k-th prime to remain legible; sign-change primes are always shown.
Chart 05SC(N) / π(N) — sign changes per primeRatio of cumulative sign changes to cumulative prime count. If O(1/N) conjecture holds, this ratio grows with N. Growth rate is an open analytic question regardless of the rate conjecture.
Chart 06Sign change gap distributionHistogram of gap sizes between consecutive sign changes. An approximately exponential tail is consistent with pseudo-random oscillation driven by primes. A heavy tail or periodicity would indicate structured interference.
Local structure & χ₄ connection
Chart 07Local lift factor φ(M+1) / [(M+1)·C]Each term L(M) pushes R(N) upward when this factor exceeds 1 (gold line), downward when below. At M = prime, φ(M) = M−1 is maximal — the teal vertical bands show prime positions. The sign of (factor − 1) is a strong asymptotic predictor of E(N)’s next direction — exact in the limit as R(N−1)→C, but with finite-N error of order |E(N−1)|.
Chart 08χ₄-restricted E(N) · gcd(M, 4) = 1E(N) restricted to M ≡ 1 or 3 (mod 4) — the support of the Dirichlet character χ₄. Scaling by ×4 (the conductor) connects this to the Farey–χ₄ discrepancy and the zeros of L(s, χ₄). Teal dots mark sign changes in this restricted sum. The Franel–Landau theorem states that the Riemann Hypothesis is equivalent to a discrepancy bound on Farey sequences; the χ₄ restriction is the simplest non-trivial Dirichlet character analogue, making its sign changes direct probes of L(s, χ₄) zero structure.
Chart 09Prime-at-Sign-Change fraction — SC at prime / total SCRunning fraction of sign changes that occur exactly at a prime N. If this fraction stabilizes, it suggests a structural prime–oscillation coupling. If it drifts, sign changes decouple from primes at scale. Whether this fraction has a finite limit is an open question.
Analytic results & conjectures — 2026
Theorem A — Convergence
R(N) = Σd sqfree μ(d)/d · S(d,N)/B(N)
S(d,N)/B(N) → d / ∏p|d(p²−1)
∴ R(N) → C = ∏p(p²−2)/(p²−1) ■
Proof sketch. Write f(m) = φ(m+1)/(m+1) = Σd|m+1 μ(d)/d (Euler’s formula).
Then Σ L(m) = Σ φ(m)·f(m) = Σd μ(d)/d · S(d,N)
where S(d,N) = Σm≡−1 mod d φ(m).
By equidistribution of φ in arithmetic progressions, S(d,N)/B(N) → d/∏p|d(p²−1).
Applying Möbius inversion gives R(N) → Σd μ(d)/∏p|d(p²−1)
= ∏p(1−1/(p²−1)) = ∏p(p²−2)/(p²−1) = C.
Conjecture B — Rate
E(N) = R(N) − C = O(1/N) (numerical conjecture)
E(N)·N is bounded and oscillating ■
Evidence. Verified numerically to N = 2,000,000: max |E(N)·N| < 0.929, regression slope = −0.998 over 200,000 sample points. Mean |E(N)·N| ≈ 0.20.
This is faster than the naive O(1/log N) prediction — the ratio structure of
R(N) = A(N)/B(N) produces cancellation that eliminates the leading 1/log N term.
Chart 02 above plots E(N)·N directly; its bounded oscillation is the visual proof.
Open problem. Prove E(N)·N is bounded analytically. This reduces to bounding
S(d,N) − d·B(N)/∏p|d(p²−1) via character sums for Dirichlet characters χ mod d,
which connects to zeros of L(s,χ). Under GRH the error in S(d,N) is O(d√N log N),
giving E(N) = O(log N / N3/2).
Note — Boikova (2025)
Boikova independently derives the √x/log x error coefficient for π(x) via sinc-function analysis
of the Sieve of Eratosthenes (Preprints.org, Dec 2025, doi:10.20944/preprints202512.0862.v1).
Both frameworks arrive at the same coefficient from entirely different directions.
The open problem in both works is the same: connect the oscillation of E(N) to the imaginary
parts of L-function zeros via an explicit formula.