Complete Computational Analysis of Error Terms in Arithmetic Lattices
Let N(R) denote a counting function associated with a k-dimensional lattice problem subject to Möbius-filtered arithmetic constraints (such as coprimality or k-free conditions). Then
where the error term is controlled by the (k−1)-dimensional boundary measure of the region.
Equivalently, when translating from lattice radius R to an integer counting parameter x ~ Rk, the error term becomes:
Thus, the critical error exponent is (k−1)/k.
The origin of the error term can be understood geometrically:
1. Interior Cancellation. Möbius inversion produces near-complete cancellation in the interior of the lattice region.
2. Boundary Truncation. Near the boundary, arithmetic divisibility constraints are only partially represented due to truncation effects.
3. Boundary Dominance. Since the boundary of a k-dimensional region scales as Rk−1, the surviving error contribution is necessarily of that order.
The error term is therefore not an analytic anomaly, but a geometric inevitability.
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The main term constant is determined by the classical identity:
which gives the asymptotic density of admissible points.
| k | ζ(k) | 1/ζ(k) | Closed Form |
|---|---|---|---|
| 2 | 1.6449340668 | 0.6079271019 | π²/6 |
| 3 | 1.2020569032 | 0.8319073725 | Apéry's constant |
| 4 | 1.0823232337 | 0.9239384669 | π⁴/90 |
| 5 | 1.0369277551 | 0.9643895748 | — |
| 6 | 1.0173430620 | 0.9829523809 | π⁶/945 |
This framework explains several well-known results:
| Structure | Dimension | Main Term | Error Term |
|---|---|---|---|
| Squarefree integers | k = 2 | 6x/π² | O(x1/2) |
| Cubefree integers | k = 3 | x/ζ(3) | O(x1/3) |
| k-free integers | k | x/ζ(k) | O(x1/k) |
| Coprime pairs | k = 2 | 6R²/π² | O(R) |
| Coprime m-tuples | m | Rm/ζ(m) | O(Rm−1) |
In each case, the exponent arises from the codimension-one boundary of the associated Möbius-filtered lattice.
| Quantity | Value | Notes |
|---|---|---|
| Predicted (Main Term) | — | — |
| Actual Count | — | Exact enumeration |
| Error | — | Actual − Predicted |
| Error Bound O(Rk−1) | — | — |
| Relative Error | — | |Error|/Predicted |
| |Error|/Bound Ratio | — | — |
Testing multiple shell thicknesses to verify error concentration at boundary:
| Shell δ | Range | Boundary Pts | Interior Pts | Boundary % | Bnd Density | Int Density |
|---|
| Radius Range | Total | Surviving | Density | Deviation |
|---|
Full range computation of error vs theoretical bound O(Rk−1):
| R | Actual | Predicted | Error | |Error| | Bound | Ratio |
|---|
Verifying Σ μ(d)/dk → 1/ζ(k):
| Truncation N | Partial Sum | Target | Error | Rel Error |
|---|
Same radius across dimensions to verify exponent dependence:
| k | Exponent | Total | Surviving | Predicted | Error | Bound | Ratio |
|---|
| gcd | Count | Percentage | μ(gcd) |
|---|
This principle does not assert new analytic bounds, nor does it rely on conjectures such as the Riemann Hypothesis. Rather, it provides a geometric explanation for why known error exponents take their observed values across a wide class of arithmetic counting problems.
The Boundary Cancellation Principle should therefore be viewed as an interpretive framework, unifying classical sieve results through boundary geometry.