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Number Theory

Number Theory

The following section presupposes the chapters on Set Theory and Nonstandard Analysis. Let \(m, n \in {}^{\omega}\mathbb{N}\) and \(k \in \mathbb{N}\).

Bounding theorem for \(\omega\)-transcendental numbers: Every non-zero complex number whose imaginary or real part has absolute value is \(\le \hat{\omega}\) or \(\ge \omega\) is automatically \(\omega\)-transcendental.

Proof: In a polynomial or series equation, set \({a}_{m} = 1\) and \({a}_{k} = -{\acute{\omega}}\) for \(k < m\), then the claim in the real case follows from the geometric series formula after taking the reciprocal. We can find the exact limit value by replacing \(\omega\) by \({\omega}(m) = \omega - \acute{\omega}/{\omega(m)}^{m}\). The complex cases are solved by setting \(x = (1 + ib)\omega\) for \(b \in {}^{\omega }{\mathbb{R}}\).\(\square\)

Coefficient theorem for \(\omega\)-transcendental numbers: Every normalised irreducible polynomial and series such that that \(|{a}_{k}| \ge \omega\) for at least one \({a}_{k}\) only has \(\omega\)-transcendental zeros.

Proof: The zeros of normalised irreducible polynomials and series are pairwise distinct and uniquely determined. Since they are not \(\omega\)-algebraic, they must be \(\omega\)-transcendental.\(\square\)

Approximation theorem for \(\omega\)-algebraic numbers: Every real \(\omega\)-algebraic number of degree \(n > 1\) may be approximated by a real \(\omega\)-algebraic number of degree \(e^\nu := m < n\) with an average asymptotic error of \(\hat{\nu}\iota \zeta(\grave{m}){|{}^{\omega }\mathbb{Z}|}^{-m}\).

Proof: On the conventionally real axis, the number of \(\omega\)-algebraic numbers approximately evenly distributed between fixed limits increases by a factor of approximately \(|{}^{\omega }\mathbb{Z}|\) per degree. The error corresponds to the distance between \(\omega\)-algebraic numbers. The non-real \(\omega\)-algebraic numbers are less dense.\(\square\)

Conclusion: Two distinct real \(\omega\)-algebraic numbers have an average distance of at least \(\hat{\varphi}\pi{|{}^{\omega }\mathbb{Z}|}^{-\acute{\omega}}\). Determining this minimum distance exactly requires an infinite non-linear non-convex optimisation problem to be solved. Therefore, the \(c\)-algebraic numbers have an approximate order of \(\mathcal{O}(c)\). This disproves Roth's theorem, which does not prove more than the (trivial) minimum distance between two rational numbers. Thus, the abc conjecture is wrong, but not Liouville's result.

Theorem: The maximum distance between two neighbouring real \(\omega\)-algebraic numbers is \(\Omega/\acute{\omega}\) for the \(\omega\)-transcendental omega constant \(\Omega = e^{-\Omega} = W(1)\) (see below Lambert-W function).

Proof: The distance between two real \(\omega\)-algebraic numbers is largest around the points \(\pm 1\). The number 1 may be approximated by an real \(\omega\)-algebraic x that satisfies the polynomial or series equation \(\acute{x}x^{\acute{m}}\acute{\omega} = 1\) for \(x > 1\) or \(x^m = -\acute{x}\acute{\omega}\) for \(x < 1.\square\)

Theorem: For every number \(z \in \mathbb{Q}+i\mathbb{Q}\) that is neither 0 nor root of unity, the geometric series \(\sum\limits_{n=0}^{\omega}{{{z}^{n}}}=\widehat{1-z}(1-{{z}^{\grave{\omega}}})\) is already \(\omega\)-transcendental.

Proof: The modulus of either the numerator or denominator \({z}^{\grave{\omega}}\) is \(>{2}^{\omega/2}.\square \)

Theorem: Euler's number \(e\) is \(\omega\)-transcendental.

Proof: If we accept the exponential series as a representation of \(e\), it follows that \(e = (k{\omega} + 1)/\omega!\) for \(k > \omega\). Therefore, the numerator and the denominator of this fraction must be \(> \omega\), since neither \(\omega\) nor a prime divisor of \(k\) in the numerator simplifies with \(\omega!\). However, if we accept the representation \({(1 + \hat{\omega})}^{\omega}\) for \(e\), the claim is trivial. Note that these two representations give different numbers.\(\square\)

The greatest-prime criterion for \(\omega\)-transcendental numbers: If a real number may be represented as an irreducible fraction \(\widehat{ap}b \pm \hat{s}t\), where \(a, b, s\), and \(t\) are natural numbers, \(abst \ne 0\), \(a + s > 2\), and the (second-)greatest prime number \(p \in {}^{\omega }\mathbb{P}, p \nmid b\) and \(p \nmid s\), then \(r\) is \(\omega\)-transcendental.

Proof: The denominator \(\widehat{ap s} (bs \pm apt)\) is \(\ge 2p \ge 2\omega - \mathcal{O}(\ell) > \omega\) by the prime number theorem.\(\square \)

Theorem: Pi \(\pi\) is \(\omega\)-transcendental.

Proof: This follows from its Wallis product representation, or its product representation using the gamma function with value \(-\hat{2}\), provided that we accept these representations. It should be noted that these two representations yield distinct numbers. Alternatively, we can apply the greatest-prime criterion to the Leibniz series, or the Taylor series of arcsin\((x)\) at \(x = 1\).\(\square\)

Theorem: The constants of \(({C}_{Artin})\), Baxter \(({C}_{2})\), Chaitin \(({\Omega}_{F})\), Champernowne \(({C}_{10})\), Copeland-Erdős \(({C}_{CE})\), Erd\H{o}s-Borwein \((E)\), Feller-Tornier \(({C}_{FT})\), Flajolet and Richmond \((Q)\), Glaisher-Kinkelin \((A)\), Heath-Brown-Moroz \(({C}_{HBM})\), Landau-Ramanujan \((K)\), Liouville \(({£}_{Li})\), Murata \(({C}_{M})\), Pell \(({P}_{Pell})\), Prouhet-Thue-Morse \((\tau)\), Sarnak \(({C}_{sa})\) and Stephen \(({C}_{S})\) as well as the Euler resp. Landau totient constant \((ET\) resp. \(LT)\), the twin prime constant \(({C}_{2})\) and the carefree constants \(({K}_{1}, {K}_{2}\) and \({K}_{3})\) are \(\omega\)-transcendental, since an existing (large) power of a small or very large prime cannot be removed from numerator or denominator by simplifying.\(\square\)

Remark: The claim for \({C}_{CE}\) clearly also holds for every base from \({}^{c}\mathbb{N}^{*}\).

Theorem: The constants of Catalan \((G)\), Gieseking \((\pi \, \ln \, \beta)\), Smarandache \(({S}_{1})\) and Taniguchi \(({C}_{T})\) are \(\omega\)-transcendental because of the greatest-prime criterion.\(\square\)

Theorem: The trigonometric and hyperbolic functions and their inverse functions, the digamma function \(\psi\), the Lambert-\(W\)-function, the \(Ein\) function, the (hyperbolic) sine integral \(S(h)i\), the Euler's Beta function \(B\), and, for positive natural numbers \(s\) and \(u\) and natural numbers \(t\), the generalised error function \({E}_{t}\), the hypergeometric function \({}_{0}{F}_{t}\), the Fresnel integrals \(C\) and \(S\) and the Bessel function \({I}_{t}\) and the Bessel function of the first kind \({J}_{t}\), the Legendre function \({\chi}_{t}\), the polygamma function \({\psi}_{t}\), the generalised Mittag-Leffler function \({E}_{s,t}\), the Dirichlet series \(\sum\limits_{n=1}^{\omega}{{\hat{n}^{s}f(n)}\;}\) with maximally finite rational \(|f(n)|\), the prime zeta function \(P(s)\), the polylogarithm \({Li}_{s}\) and the Lerch zeta-function \(\Phi(q, s, r)\) always yield \(\omega\)-transcendental values for rational arguments and maximal finite rational \(|q|\) and \(|r|\) at points where their Taylor series converge.

Proof: The claim follows from the greatest-prime criterion, the Dirichlet prime number theorem, and the Wallis product. For the digamma function, the claim follows from the proof of \(\omega\)-transcendence of Euler's constant below.\(\square\)

Theorem: The gamma function \(\Gamma(z) := k! \, {k}^{z}/(z(z + 1) ... (z + k))\), where \(k = {\omega}^{{\omega}^{2}}\) and \(z \in {}^{\omega }\mathbb{C} \setminus -{}^{\omega }\mathbb{N}\), is \(\omega\)-transcendental for \(z \in {}^{\omega }\mathbb{Q}\) and for suitable supersets of \({}^{\omega }\mathbb{N}\) resp. \({}^{\omega }\mathbb{Q}\).

Proof: The values of \(\Gamma(z)\) are the zeros of minimal polynomials or series with infinite integer coefficients.\(\square\)

Theorem: For \(x \in {}^{\omega }{\mathbb{R}}\), let be \(s(x) := \sum\limits_{n=1}^{\omega}{\hat{n}{{x}^{n}}}\) and \(\gamma := s(1) - \ln \, \omega = \int\limits_{1}^{\omega}{\left( \widehat{\left\lfloor x \right\rfloor} - \hat{x} \right)dx} \in \; ]0, 1[\) (can be seen by rearranging) Euler's constant. If we accept \(s(\hat{2})\ell\) as a representation of ln \(\omega, \gamma\) is therefore with a precision \(\mathcal{O}({2}^{-\omega}\hat{\omega}\ell)\) \(\omega\)-transcendental.

Proof: We obtain \(-\ln(-\acute{x}) = s(x) + \mathcal{O}(\hat{\omega}{x}^{\grave{\omega}}/\acute{x}) + t(x)dx\) for \(x \in [-1, 1 - \hat{c}]\) and a real function \(t(x)\) such that \(|t(x)| < {\omega}\) by (exact) integration (see Nonstandard Analysis) of the geometric series. After applying Fermat's little theorem to the numerator of \(\hat{p}(1 - 2^{-p}\ell)\) for \(p = \max\, {}^{\omega}\mathbb{P}\), the greatest-prime criterion yields the claim.\(\square\)

Remark: If \(\omega\) is replaced by an arbitrary \(k \in {}^{\omega }\mathbb{N}_{\ge\omega/2}\), the preceding proof is barely more difficult.

Definition: When two numbers \(x, y \in {}^{\omega }\mathbb{C}^{*}\) or their reciprocals do not satisfy any polynomial or series equation \(p(x, y) = 0\), so they are called \(\omega\)-algebraically independent.

Theorem: The greatest-prime criterion, with \(e = {(1 + \hat{p})}^{p}\) for maximal \(p \in {}^{\omega }\mathbb{P}\) and \(\pi\) as Wallis product, yields pairwise \(\omega\)-algebraically independent representations of \(A, {C}_{2}, \gamma, e, K\) and \(\pi.\square\)

Theorem: The BBP series \(\sum\limits_{n=1}^{\omega}{p(n)\widehat{q(n){{b}^{n}}}}\) for \(b \in {}^{\omega }\mathbb{N}_{\ge 2}\) and integer polynomials resp. series \(p\) and \(q \in {}^{\omega }\mathbb{Z}\) with \(q(n) \ne 0\) and \(\deg(p) < \deg(q)\) only yield \(\omega\)-transcendental values.

Proof: We can reduce the sum to a smallest common denominator \(d \ge {b}^{k} > \omega\) with \(d, k \in \mathbb{N}^{*}.\square\)

Definition: A rational number \(\ne 0\) is said to be power-free if it cannot be represented as the power of a rational number with integer exponent \(\ne \pm 1\). Let \(||\cdot|{{|}_{d}}\) be the distance to the nearest integer.

Theorem: For any power-free \(q \in Q := {\mathbb{Q}}_{>0}\), we have that \({q}^{x} \in Q\) for real \(x\) if and only if \(x \in {}^{\omega }\mathbb{Z}\) and \(|x|\) is not excessively large.

Proof: Let wlog \(x > 0\). Since there is no contradiction for \(x \in {}^{\omega }\mathbb{N}\), assume \(x \in Q \setminus {}^{\omega }{\mathbb{N}}^{*}\). Since this implies \({q}^{x} \in {}^{\omega }{\mathbb{A}}_{R} \setminus Q\), assume \(x := k/d \in {}^{\omega }\mathbb{R}_{>0} \setminus Q\) for \(d, k \in {\mathbb{N}}^{*}\) and gcd\((d, k) = 1\). This implies \({q}^{k} = {r}^{d}\) for an \(r \in Q\). The fundamental theorem of arithmetic yields a numerator or denominator of \(q\) or \(r\) greater than \(2^{\omega}\). This contradiction results in the claim.\(\square\)

Remark: This theorem proves the Alaoglu and Erdős conjecture, which states that \({p}^{x}\) and \({q}^{x}\) are \(c\)-rational for distinct \(p, q \in {}^{c}\mathbb{P}\) if and only if \(x \in {}^{c}\mathbb{Z}\) and \(|x|\) is not excessively large. By replacing \({}^{c}\mathbb{N}\) with \({}^{\omega }\mathbb{N}\) and making the required adjustments these arguments can be extended to finite transcendental numbers. Inconcrete transcendence implies finite transcendence.

Littlewood theorem in conventional mathematics: We have for all \(a,b\in {}^{c}\mathbb{R}\) and \(n\in {}^{c}\mathbb{N}^{*}\):\[\underset{n\to \infty }{\mathop{\lim \inf }}\,n\;||na|{{|}_{d}}\;||nb|{{|}_{d}}=0.\]Proof: Let be \(k, m \in {}^{c}\mathbb{N}^{*}\) the denominators of the continued fraction of \(a\) resp. \(b\) with precision \(g \in {}^{c}\mathbb{R}_{> 0}\) and \(n\) again and again a natural multiple of \(km\). Then we have according to Dirichlet's approximation theorem (see \cite{455}, p. 63): \[\underset{n\to \infty }{\mathop{\lim \inf }}\,n||na|{{|}_{d}}||nb|{{|}_{d}}=\underset{n\to \infty }{\mathop{\lim \inf }}\,n\mathcal{O}{{(\hat{n})}^{2}}=\underset{n\to \infty }{\mathop{\lim \inf }}\,\mathcal{O}(\hat{n})=0.\square\]Refutation of the Littlewood conjecture in nonstandard mathematics: Let \(a = b := {{\omega}^{-{3}/{2}}}\). Then we have: \[\omega \;||\omega a|{{|}_{d}}\;||\omega b|{{|}_{d}}= 1 \ne 0.\square\]Theorem: The generalised Riemann hypothesis is disproved by the Dirichlet \(L\)-function \(L\left(s,\chi\right)=\sum\limits_{n=1}^{\omega}{\chi\left(n\right)n^{-s}}\), which clearly has because of the geometric series (cf. Set Theory) only zeros for \(s = 0\) and nontrivial Dirichlet characters \(\chi(n)\).\(\square\)

© 2009-2018 by Boris Haase

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