The implied constant in E(N) = O(1/N) appears to equal the product of the two landmark values of the analytic lift function F(u) at u = 0 and u = 2. Its maxima occur exclusively at N = p−1 for primes p, with the highest values governed by safe primes p = 2q+1.
The Lift Survival Ratio R(N) = ΣL(m)/Σφ(m) converges to C = ∏p(p²−2)/(p²−1) ≈ 0.530711806246. The error E(N) = R(N) − C is conjectured to satisfy E(N) = O(1/N), meaning the product |E(N)·N| remains bounded for all N. K is the tightest such bound — the supremum of |E(N)·N| over all N ≥ 2.
| N | |E(N)|·N | vs ζ(2)C | N+1 prime | Safe prime | Type |
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By the exact step formula (Lemma 1 of the main conjecture paper), E(N) decreases at every prime p ≥ 3 because φ(p+1)/(p+1) ≤ 1/2 < C ≈ 0.531. This means E(N) is at a local maximum immediately before the prime arrives — at N = p−1. All 17 events with |E·N| > 0.85 in the first million steps occur at such positions, with zero exceptions.
When p−1 = 2q with q prime, the lift factor at step p−1 equals φ(2q)/(2q) = (q−1)/(2q), which approaches 1/2 from below as q grows. Since 1/2 is the minimum possible lift factor for an even number, this creates the cleanest possible buildup of E before the prime drop. All top-20 maxima of |E·N| occur at primes with φ(p−1)/(p−1) very close to 1/2, with safe primes accounting for the most extreme values.
The implied constant in E(N) = O(1/N) is the product of F(u) evaluated at its two non-trivial landmark values u = 0 and u = 2. F(1) = 1 exactly is the geometric pivot: K = F(0)·F(2)/F(1)². The values above ζ(2)·C observed numerically (11 out of 2,000,000 steps) appear to be finite transients — small N where the Euler product for C has not yet stabilized.
The conjecture K = ζ(2)·C is a conjecture within a conjecture: it asserts not just that
E(N) = O(1/N) (the main conjecture), but that the implied constant is exactly F(0)·F(2).
The 11 observed exceedances all occur at N < 1,300,000 and the excess is < 3%.
Whether these are permanent exceedances (meaning K > ζ(2)·C) or transients
that eventually fall below is the key open question.
A weaker but provable statement: among safe primes p = 2q+1,
|E(p−1)|·(p−1) appears to approach ζ(2)·C from below as q → ∞
through Sophie Germain primes. This is a conditional statement about the density of
large safe primes — again unproved but numerically strong.
F(u) is strictly decreasing on [0, 4) with three distinguished values: F(0) = ζ(2) (density of all squarefrees), F(1) = 1 (exact, trivial), F(2) = C (lift survival constant). The conjecture K = F(0)·F(2) says the implied constant is the product of the outer two landmarks. Since F(1) = 1 exactly, this equals F(0)·F(2)/F(1)² = F(0)·F(2) — the product of the deviations of F from 1 on each side. Geometrically: K = (F(0)−1+1)·(1−(1−F(2))) = how far F extends above 1 at u=0, times how far it sits below 1 at u=2, measured as a product rather than a sum.
Are the 11 observed exceedances permanent (making K > ζ(2)·C) or transients that eventually disappear? This requires either proving the bound analytically or finding an N where the ratio |E(N)|·N/(ζ(2)·C) exceeds 1 arbitrarily — a definitive numerical test at N = 109 or beyond would be informative.
Does limq→∞, q Sophie Germain prime |E(2q)|·2q equal ζ(2)·C? If yes this would be a conditional theorem (conditional on infinitely many Sophie Germain primes, which is itself unproved). The numerical evidence strongly suggests it — every safe prime up to N = 1,000,000 gives a value below ζ(2)·C except p = 289,559.
Is there a direct derivation of K = F(0)·F(2) from the structure of the Dirichlet series for ΣL(m)m−s? The Perron integral approach would express E(N) as a contour integral whose residues involve F(u) evaluated at specific points. Whether the residue at the leading pole equals ζ(2)·C is the key analytic question.
Under GRH, E(N) = O(log N / N3/2), which means E(N)·N → 0. In that case K = lim sup E(N)·N would equal zero, not ζ(2)·C. So if K = ζ(2)·C holds, the O(1/N) conjecture would be exactly tight (not improvable to o(1/N)) without GRH, and GRH would provide a strictly better rate. This interplay between the two rates is unexplored.