# Homepage of Boris Haase ## 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 $$x = \grave{y}\omega$$ 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 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 $$\Gamma(z) := k! \, {k}^{z}/(z\grave{z} ... (z + k))$$, 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: 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 $$\chi(n)$$.$$\square$$