Comprehensive analysis with RH connections, modular arithmetic, and advanced visualizations
Asymptotic count of coprime pairs (r, m) with r/m in sector S_n = (1/(n+1), 1/n]
Definition: Sectors are defined by consecutive Farey bounds: sector S_n spans the interval (1/(n+1), 1/n], a decreasing sequence of intervals that partition [0,1]. Formula Purpose: This asymptotic formula provides increasingly accurate predictions of the count of coprime pairs (r,m) with r/m in sector S_n, where N is the maximum denominator. As N grows, the accuracy of this prediction improves significantly. The formula shows how the Farey sequence distributes its fractions across the unit interval with mathematical precision.
| n | Interval | Exact | Pred | Err% | Primes | Composites |
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Wessen Getachew
We study the distribution of coprime integer pairs (a,b) with bounded height whose rational slope lies in a fixed Farey sector S_n = (1/(n+1), 1/n]. Using classical summatory totient estimates together with a geometric decomposition of rational directions, we derive an explicit asymptotic formula for the number of primitive lattice points in each sector. This result provides a localized refinement of the global coprime density 1/zeta(2), revealing directional structure in the distribution of visible lattice points.
The probability that two randomly chosen integers are coprime is 1/zeta(2) = 6/pi^2 approximately 0.6079, a classical result with interpretations in analytic number theory and geometry of numbers. Geometrically, this corresponds to the density of visible lattice points in Z^2.
Farey sequences organize rational numbers in [0,1] by increasing denominator and naturally partition rational directions into intervals. While global coprime density is well understood, we focus on directional coprime density - how primitive lattice points distribute across specific rational slope bands.
Intervals partitioning (0,1], bounded by consecutive Farey fractions.
Pairs with coprime coordinates representing visible lattice points.
Let n be fixed. As N approaches infinity, the number of coprime integer pairs with slope in the Farey sector S_n satisfies:
Where C(n,N) counts pairs (a,b) with 1 <= b <= N, gcd(a,b)=1, and a/b in S_n.
Step 1: The condition a/b in S_n = (1/(n+1), 1/n] is equivalent to b/(n+1) < a <= b/n.
Step 2: For each b, the number of integers a in this range is b/(n(n+1)) + O(1).
Step 3: Restricting to coprime pairs: Sum over b<=N of phi(b)/(n(n+1)) + O(N)
Step 4: Apply the classical summatory totient estimate:
Conclusion: C(n,N) = (1/(n(n+1))) * ((3/pi^2)N^2 + O(N log N))
| Sector n | Width 1/(n(n+1)) | Asymptotic Factor | Global Fraction |
|---|---|---|---|
| 1 | 1/2 | 3/(2pi^2) | approximately 0.1519 |
| 2 | 1/6 | 3/(6pi^2) | approximately 0.0506 |
| 3 | 1/12 | 3/(12pi^2) | approximately 0.0253 |
| n | 1/(n(n+1)) | 3/(pi^2 n(n+1)) | 1/n(n+1) * 6/pi^2 |
Numerical experiments confirm convergence to predicted asymptotics at rates consistent with O(N log N) error decay. Interactive visualizations above display accumulation of primitive lattice points within each Farey sector.
Global density 1/zeta(2) = 6/pi^2 emerges as sum of sectoral contributions.
Farey gap distribution relates to RH via: Sum|F_k - k/|F_N|| = O(N^(1/2+epsilon)) iff RH
Farey sectors provide a natural geometric decomposition of rational directions. By combining this structure with classical coprime density results, we obtain an explicit directional refinement of visible lattice point counts. This framework enables anisotropic analysis of arithmetic distributions and provides foundation for modular constraints, higher-dimensional generalizations, and computational number theory.
Acknowledgments: Foundational work of Euler, Mertens, and others on totient summation and Farey geometry.
Visual exploration of the Farey Sector Density Theorem with interactive elements
For consecutive Farey fractions a/b and c/d, the gap between them equals exactly 1/(b*d). This elegant formula shows that gaps decrease as denominators grow. The distribution reveals how fractions cluster and spread across the unit interval.
| # | Left Fraction | Right Fraction | Actual Gap | Theoretical 1/(bd) | Match |
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Define δ_k = F_k - k/|F_N| as the k-th Farey fraction deviation. The growth rate of Σ|δ_k| is directly equivalent to the Riemann Hypothesis. If RH is true, the sum grows slower than N^(1/2+ε) for any ε > 0. This provides a concrete computational test for RH properties through Farey sequence analysis. Note: These computations illustrate known equivalences but do not constitute a proof or disproof of RH.
Dedekind sums are arithmetic functions encoding properties of coprime pairs (h,k). They satisfy reciprocity formulas: s(h,k) + s(k,h) = -1/4 + (1/12)(h/k + k/h + 1/(hk)). These sums connect to modular forms and are fundamental in algebraic number theory.
| h | k | s(h,k) | 12k·s(h,k) | gcd |
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| r/m | CF [a0; a1, a2, ...] | Path | Length | Sum(a_i) | Freq | Play |
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| r/m | θ (rad) | R | z = Re^iθ | Γ = (z-1)/(z+1) | |Γ| |
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Map Farey fractions to musical frequencies, explore consonance/dissonance, and visualize Arnold tongues
| Fraction | Ratio | Frequency | Cents | Closest Note | Interval Name | Mode | Play |
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Exploring how primorial residue classes connect to Farey sector distribution
| Sector | Total | Per Class Avg | χ² Stat | Uniform? |
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| N | |F_N| | Σ|δ| | Σ|δ|/√N | Max Gap | Mean Gap |
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