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Number Theory

Number Theory

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

Minimality theorem: For the unique \(b\)-adic expansion of every \(r \in \mathbb{R}\), min \(\{b \in \mathbb{R}_{>1}\} = 2\) holds.

Proof: Focusing on the positions after the floating point, the geometric series yields the claim.\(\square\)

Prime number theorem: For \(\pi(x) := |\{p \in {}^{\omega}{\mathbb{P}} : p \le x \in {}^{\omega}{\mathbb{R}}\}|\) holds \(\pi(\omega) = \omega/\ln \omega + \mathcal{O}(\sqrt{\omega} \ln \omega)\).

Proof: The sieve of Eratosthenes shows that the number of increasing prime numbers decreases regularly with negligible deviations. From intervals of fix length \(y \in {}^{\omega}{\mathbb{R}_{>0}}, \hat{2}y\) set-2-tuples of prime numbers are formed such that the first interval has the unchanged representative prime number density and the second interval is empty, then the interval with the second most prime number density is followed by the second least one etc. For induction basis \(n = 2\) resp. 3, the induction hypothesis is that the first interval contains \(x_n/\ln\,x_n\) prime numbers for \(n \in {}^{\omega}{\mathbb{N}_{\ge2}}\) and arbitrary \(x_4 \in [2, 4[\). Then the induction step from \(x_n\) to \(x_n^2\) proves that there are \(\hat{2}x_nx_n/\ln\,x_n = x_n^2/\ln\,x_n^2\) prime numbers. The average distance between the prime numbers is \(\ln\,x_n\) and \(x_n^2\) to \(x_n\) behaves like \(\omega\) to \(\sqrt{\omega}.\square\)

Remark: Replacing the number 2 by \(m \in {}^{\omega }{\mathbb{N}_{>2}}\) for \(\hat{m}{y}^{\acute{m}}\) set-\(m\)-tuples gives the same result.

Theorem: The differences \(q - p\) for \(p, q \in {}^{\omega }{\mathbb{P}_{\ge3}}\) generate even integers without gaps.

Proof: By induction using Bertrand's postulate.\(\square\)

Goldbach’s theorem: The sums \(r + q\) for \(q, r \in {}^{\omega }{\mathbb{P}_{\ge3}}\) generate whole \({2}^{\omega }{\mathbb{N}_{\ge3}}\).

Proof: For \(p, q, r \in {}^{\omega }{\mathbb{P}_{\ge3}}\), the \(|p \pm s|\) generate whole \({2}^{\omega }{\mathbb{N}_{\ge3}}\) after \(s := \pm r \pm q \mp p\) do so whole \({2}^{\omega }{\mathbb{Z}}\mp3.\square\)

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. The exact limit value can be found by replacing \(\omega\) by \({\omega}(m) = \omega - \acute{\omega}/{\omega(m)}^{m}\). The complex cases are solved by setting \(x = \grave{y}\omega\) for \(y \in i{}^{\omega }{\mathbb{R}^{*}}.\square\)

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 \(\sum\limits_{n=0}^{\omega}{{{z}^{n}}}=({{z}^{\grave{\omega}}} - 1)/\acute{z}\) 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 the exponential series is accepted 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 the representation \({(1 + \hat{\omega})}^{\omega}\) is accepted 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 \(\ge 2p \ge 2\omega - \mathcal{O}(\ell\sqrt{\omega}) > \omega\) 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 these representations are accepted. It should be noted that these two representations yield distinct numbers. Alternatively, the greatest-prime criterion can be applied 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 \(({C}_{CE})\), Erdős-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 \(\Gamma(z) := k! \, {k}^{z}/(z\grave{z} ... (z + k))\), 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: For \(x := \Gamma(z), a_n \in \mathbb{Z}\) und \(\sum\limits_{n=0}^{\omega}{a_n{{x}^{n}}} = 0\), there is at least one \(n\) such that \(|a_n| > \omega.\square\)

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

Proof: The (exact) integration of the geometric series (see Nonstandard Analysis) yields \(-\ln(-\acute{x}) = s(x) + \mathcal{O}(\hat{\omega}{x}^{\grave{\omega}}/\acute{x}) + t(x)dx\) for \(x \in [-1, 1 - \hat{c}]\) and a real function \(t(x)\) such that \(|t(x)| < {\omega}\). 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.\(\triangle\)

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: 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.\(\triangle\)

Theorem: If all \(q \in Q := {\mathbb{Q}}_{>0}\) are power-free, \({q}^{x} \in Q\) and \(\ell \gg |x| \in {}^{\omega}\mathbb{R}\), it must \(x \in {}^{\omega }\mathbb{Z}\) hold.

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: For all \(a,b\in {}^{c}\mathbb{R}\) and \(n\in {}^{c}\mathbb{N}^{*}\), it holds that:\[\underset{n\to \infty }{\mathop{\lim \inf }}\,n\;||na|{{|}_{d}}\;||nb|{{|}_{d}}=0.\]Proof: For \(k, m \in {}^{c}\mathbb{N}^{*}\) as denominators of the continued fraction of \(a\) resp. \(b\) with precision \(g \in {}^{c}{\mathbb{R}}_{>0}\) and \(n/km\) again and again integer, Dirichlet's approximation theorem (see [455], p. 63) yields that: \[\underset{n\to \infty }{\mathop{\lim \inf }}\,n||na|{{|}_{d}}||nb|{{|}_{d}}=\underset{n\to \infty }{\mathop{\lim \inf }}\,n\mathcal{O}{{(\hat{n})}^{2}}=\underset{n\to \infty }{\mathop{\lim \inf }}\,\mathcal{O}(\hat{n})=0.\square\]Refutation of the Littlewood conjecture in nonstandard mathematics: Let \(a = b := {{\omega}^{-{3}/{2}}}\). Then it holds that: \[\omega \;||\omega a|{{|}_{d}}\;||\omega b|{{|}_{d}}= 1 \ne 0.\square\]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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