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#34: Completion Number Theory on 07.06.2011

Definition: A complex number x is called downward (upward) finitely bounded if there is a finitely algebraic real number c > 0 with c ≤ |x| (|x| ≤ c). A downward and upward finitely bounded complex number is called two-sided finitely bounded.

Sum proposition for finitely transcendental numbers: Every sum a|ℕ|r + v with a two-sided finitely bounded complex a, a downward finitely bounded real r and a finitely algebraic complex v is finitely transcendental.

Proof: Inserting a|ℕ|r + v into each equation of a minimal polynomial yields in each case bO(|ℕ|s) + w ≠ 0 according to the corrected axiom of Archimedes (cf. set theory) with a two-sided finitely bounded complex b, a downward finitely bounded real s and a finitely algebraic complex w.

To these finitely transcendental numbers belongs in particular, in each case, the last sequence member that comes closer to a finitely algebraic complex number than any finitely algebraic complex number, such as the number e comes so close to the number -1. The sum proposition may still be sharpened by finitely continuing to logarithmise |ℕ|r to a two-sided finitely bounded positive real base.

Proposition: The twin prime constant C2, as product over all (1 - 1/(p - 1)2) with prime p > 2, is transcendental.

Proof: According to the prime number theorem, the greatest and second greatest prime number - both = |ℕ| - |O(log |ℕ|)| - do not divide the denominator of C2, from which the assertion follows.

Proposition: The Landau-Ramanujan constant K, as product, divided by √2, over all (1 - 1/p2) with prime p ≡ 3 mod 4, is transcendental.

Proof: According to the (Dirichlet) prime number theorem, the greatest and second greatest prime number ≡ 3 mod 4 - both = |ℕ| - |O(log |ℕ|)| - do not divide the denominator of K, from which the assertion follows.

Proposition: The Glaisher-Kinkelin constant A = 11 22 33 ... |ℕ||ℕ| (1+1/|ℕ|)|ℕ|/4/|ℕ|(|ℕ|/2+|ℕ|/2+1/12) is transcendental.

Proof: After cancelling, the greatest prime number of ℕ remains in the denominator with exponent > 2.

Greatest-prime criterion for transcendental numbers: If a real number r has, for cancelled fractions, the representation a/(bp) ± s/t with (trans-) natural a, b, s and t, abst ≠ 0 and b + t > 2, as well as the (second) greatest (trans-) natural prime number p of an index set I ⊇ ℕ with p ∤ a and p ∤ t, so it is transcendental.

Proof: It applies r = (at ± bps)/(bpt) with a denominator ≥ 2p ≥ 2|ℕ| - |O(log |ℕ|)| > |ℕ| + 1 as consequence of the prime number theorem, from which the assertion follows.

Proposition: The circular constant π is transcendental.

Proof: This follows from Wallis' product or the product representation of the gamma function for the value -½. Notice that both representations define different numbers. Alternatively, apply the greatest-prime criterion to the Leibniz series or to the arcsin(x) Taylor series for x = 1.

Proposition: The Dirichlet beta, eta and lambda function, as well as the Riemann zeta function, yield for natural arguments greater than 1 only transcendental values.

Proof: Applying the greatest-prime criterion yields directly the assertion.

Proposition: Let be s(x) for complex x the sum of xk/k over all k ∈ I. If one defines Euler's constant as γ = s(1) - log |ℕ| with I = ℕ, so it is transcendental.

Proof: We have γ = s(1) - s(1 - 1/|ℕ|) + r dx, with an I that equals the maximally possible trans-natural set, for s(1 - 1/|ℕ|), and at most trans-rational s(1) - s(1 - 1/|ℕ|), as well as real r and x. This results from the geometric series by (exact) integration over x (see nonstandardanalysis). Here s(1 - 1/|ℕ|) does not tend to log |ℕ|, because of 1/e = (1 - 1/|ℕ|)|ℕ|, before a trans-natural index O(|ℕ|2). The assertion follows from the greatest-prime criterion, provided r dx = 0, otherwise from the product and sum proposition, since dx is so small that γ is transcendental.

© 07.06.2011 by Boris Haase


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