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Number Theory

Number Theory

The following section presupposes the results established in the chapters on Set Theory and Nonstandard Analysis.

Definition: ||·||d is the distance to the next integer.

Littlewood theorem in conventional mathematics: For all a, b ∈ cℝ and n ∈ cℕ*, we have that:

formula_001

Proof: Let r and s be the denominators of a and b with precision w and n, all natural multiples of rs. Then by the Dirichlet approximation theorem:

formula_002

Refutation of the Littlewood conjecture in nonstandard mathematics: Let a = b := ώ-3/2. Then:

ώ ||ώa||d ||ώb||d = 1 ≠ 0.⃞

The generalised Riemann hypothesis holds as

Theorem: For minimal ε ∈ [½, 1], σ(0) := χ(0) = 0, σ(x) := σ(n) = ρ(n) + σ(n - 1), d := σ(ώ)/(ώ + 1)s, σ(x) = O(xε), ρ(n) = ±χ(n), an arbitrary Dirichlet character χ(n) with ⌊x⌋ = n and n ∈ ωℕ*, the Dirichlet L-function L(s, χ) with s ∈ ωℂ, x ∈ ω≥1 and

formula_005

ε = ½ holds (see [887], p. 56 f.).

Indirect proof: Assume ε ∈ ]½, 1]. If s := ½ + it with t ∈ cℝ is a non-trivial zero of L(s, χ), then also every δ + it is one with δ ∈ ]½, ε]. This yields the contradiction.⃞

Remark: The Riemann hypothesis follows from χ(n) = 1 for all n ∈ ωℕ*. Also χ2(n) is Dirichlet character. Note the functional equation for ε ∈ [0, ½[ (see [887], p. 108).

Prime number theorem: For x ∈ ω≥2657, we have by (Lowell Schoenfeld: Sharper Bounds for the Chebyshev Functions θ(x) and ψ(x). II; Mathematics of Computation Vol. 30, No. 134 (1976), 337 - 360) that

formula_006

Ternary Goldbach theorem: Every odd n ∈ ω>5 can be written according to (Jean-Marc Deshouillers et al.: Electronic Research Announcements of the AMS Vol. 3 (1997), 99 - 104) as the sum of three primes.⃞

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