Explicit estimates for the count of integral ideals in number fields are derived with error terms that grow much more slowly with the degree n than the standard n^{n^2} bound.
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Higher-order dualities yield ∑ μ(n) ω(n)^k / n = 0 for k ≥ 2 and conditional sums over smallest prime factor p1(n) ≡ j mod ℓ equal to zero or 1/φ(ℓ) for coprime j, ℓ and k ≥ 3.
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Explicit counting of ideals in number fields of arbitrary degree
Explicit estimates for the count of integral ideals in number fields are derived with error terms that grow much more slowly with the degree n than the standard n^{n^2} bound.
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Duality Between Prime Factors and The Prime Number Theorem For Arithmetic Progressions -- Higher Order Dualities
Higher-order dualities yield ∑ μ(n) ω(n)^k / n = 0 for k ≥ 2 and conditional sums over smallest prime factor p1(n) ≡ j mod ℓ equal to zero or 1/φ(ℓ) for coprime j, ℓ and k ≥ 3.