Homepage of Boris Haase




Previous

#73: Extension Number Theory on 17.04.2018

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 ∈ 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) = r(n) + σ(n - 1), d := σ(ώ)/(ώ + 1)s, σ(x) = O(xε), r(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 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 ∈ 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 and π(x) := |ℙ≤x|, we have that

formula_006

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) ∈ ℙ2.

Proof: By the prime number theorem, the number n may be written as the sum of two primes with probability ln-2n. If we sufficiently account for statistical imprecision with another factor of ln-2n, then the claim follows, since the theorem is known to hold for n ≤ 4 1018 and 1012 ln4n < n otherwise.⃞

Similarly, Lemoine's conjecture, which was verified for n ∈ [4, 5 108]ωℕ, is proven as

Corollary: The equation 2n - 1 = p + 2q ∈ ω>6 has always solutions p, q ∈ ωℙ.⃞

Similarly, it follows directly Polignac's conjecture as

Corollary: For every fixed k ∈ cℕ*, there are O(ώ/ln2ώ) numbers p, p + 2k ∈ ωℙ.⃞

Cramér's gap theorem: For every p ∈ ωℙ \ {max ωℙ} and q = min ω>p, q - p = O(ln2p) 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>0.⃞

Oppermann's conjecture follows directly as

Corollary: For every n ∈ ω>1, there is at least one (p, q) ∈ ℙ2 with n2 - n < p < n2 < q < n2 + n.⃞

© 17.04.2018 by Boris Haase


Valid XHTML 1.0 • disclaimer • mail@boris-haase.de • pdf-version • bibliography • subjects • definitions • statistics • php-code • rss-feed • top