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

Collatz theorem: The sequence n/2 for even n and 3n + 1 for odd n always ends at 1 for n ∈ ℕ*.

Proof: After one iteration, by (Jeffrey C. Lagarias: The 3x+1 problem and its generalizations; American Mathematical Monthly 92 (1985), 3 - 23), we obtain the expected value 3n/4 + O(1), from which the sequence cannot grow unlimitedly. By (Ivan Slapničar: There are no cycles in the 3n + 1 sequence; https://arxiv.org/pdf/1706.08399v1.pdf), only the trivial cycle 1 - 4 - 2 - 1 exists.⃞

Definition: || · ||_{d} is the *distance to the next integer*.

Littlewood theorem in conventional mathematics: For all a, b ∈ ^{c}ℝ and n ∈ ^{c}ℕ*, we have that:

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:

⃞

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) = r(n) + σ(n - 1), d := σ(ώ)/(ώ + 1)^{s}, σ(x) = O(x^{ε}), r(n) = ±χ(n), an arbitrary Dirichlet character χ(n) with ⌊x⌋ = n and n ∈ ^{ω}ℕ*, the Dirichlet L-function L(s, χ) with s ∈ ^{ω}ℂ, x ∈ ^{ω}ℝ_{≥1} and

ε = ½ holds (see [887], p. 56 f.).

Indirect proof: Assume ε ∈ ]½, 1]. If s := ½ + it with t ∈ ^{c}ℝ is an arbitrary non-trivial zero of L(s, χ), then also all δ + it with δ ∈ ]½, ε]. This yields the contradiction.⃞

Remark: The Riemann hypothesis follows from χ(n) = 1 for all n ∈ ^{ω}ℕ*. Also χ^{2}(n) is Dirichlet character. Note the functional equation for ε ∈ [0, ½[ (see [887], p. 108).

Prime number theorem: For x ∈ ^{ω}ℝ_{≥2657} and π(x) := |ℙ_{≤x}|, we have that

Proof: The claim follows 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.⃞

Goldbach's theorem: Every n ∈ 2ℕ* \ {2} may be written as n = p + q with (p, q) ∈ ℙ^{2}.

Proof: By the prime number theorem, the number n may be written as the sum of two primes with probability ln^{-2}n. If we sufficiently account for statistical imprecision with another factor of ln^{-2}n, then the claim follows, since the theorem is known to hold for n ≤ 4 10^{18} and 10^{12} ln^{4}n < n otherwise.⃞

Similarly, Lemoine's conjecture, which was verified for n ∈ [4, 5 10^{8}]^{ω}ℕ, is proven as

Corollary: The equation 2n - 1 = p + 2q ∈ ^{ω}ℕ_{>6} has always solutions p, q ∈ ^{ω}ℙ.⃞

Similarly, it follows directly Polignac's conjecture as

Corollary: For every fixed k ∈ ^{c}ℕ*, there are O(ώ/ln^{2}ώ) numbers p, p + 2k ∈ ^{ω}ℙ.⃞

Cramér's gap theorem: For every p ∈ ^{ω}ℙ \ {max ^{ω}ℙ} and q = min ^{ω}ℙ_{>p}, q - p = O(ln^{2}p) holds.

Proof: Because of the prime number theorem and the basically and sectionally uniform distribution of the prime gaps, the distance q - p is at most c ⌊ln p⌋ ⌈ln p⌉ for a small c ∈ ^{c}ℝ_{>0}.⃞

Oppermann's conjecture follows directly as

Corollary: For every n ∈ ^{ω}ℕ_{>1}, there is at least one (p, q) ∈ ℙ^{2} with n^{2} - n < p < n^{2} < q < n^{2} + n.⃞

© 17.04.2018 by Boris Haase

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