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= Welcome to MWiki =
 
= Welcome to MWiki =
 
== Theorems of the month ==
 
== Theorems of the month ==
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=== Definition ===
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=== Prime number theorem ===
  
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Let <math>f_n^*(z) = f(\eta_nz)</math> <em>sisters</em> of the Taylor series <math>f(z) \in \mathcal{O}(D)</math> centred on 0 on the domain <math>D \subseteq {}^{\omega}\mathbb{C}</math> where <math>m, n \in {}^{\omega}\mathbb{N}^{*}</math> and <math>\eta_n^m := i^{2^{\lceil m/n \rceil}}</math>. Then let <math>\delta_n^*f = (f - f_n^*)/2</math> the <em>halved sister distances</em> of <math>f.</math> For <math>\mu_n^m := m!n!/(m + n)!</math>, <math>\mu</math> and <math>\eta</math> form an calculus, which can be resolved on the level of Taylor series and allows an easy and finite closed representation of integrals and derivatives.<math>\triangle</math>
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For <math>\pi(x) := |\{p \in {\mathbb{P}_{\le x}} : x \in {}^{\omega}{\mathbb{R}}\}|</math>, it holds that <math>\pi(\omega) = \widetilde{{_e}\omega}\omega + \mathcal{O}({_\epsilon}\omega\;{\omega}^{\tilde{2}})</math>.
  
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=== Speedup theorem for integrals ===
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==== Proof: ====
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From intervals of fix length <math>y \in {}^{\omega}{\mathbb{R}_{>0}}, \check{y}</math> 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. The Stirling formula suggests the prime gap <math>n = {\epsilon}^{\sigma} = \mathcal{O}({_\epsilon}(n!))</math>.
  
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The Taylor series (see below) <math>f(z) \in \mathcal{O}(D)</math> centred on 0 on <math>D \subseteq {}^{\omega}\mathbb{C}</math> gives for <math>\grave{m}, n \in {}^{\omega}\mathbb{N}^*</math><div style="text-align:center;"><math>\int\limits_0^z...\int\limits_0^{\zeta_2}{f(\zeta_1)\text{d}\zeta_1\;...\;\text{d}\zeta_n} = \widehat{n!} f(z\mu_n) z^n.\square</math></div>
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For induction basis <math>n = 2</math> resp. 3, the hypothesis states the first interval to contain <math>x_n/{_\epsilon}x_n</math> primes for <math>n \in {}^{\omega}{\mathbb{N}_{\ge2}}</math> and <math>x_4 \in [2, 4[</math>. Stepping from <math>x_n</math> to <math>x_n^2</math> finds <math>\pi(x_n^2) = \pi(x_n) \check{x}_n</math> primes only from <math>\pi(x_n) = x_n/{_\epsilon}x_n</math>. The average prime gap is <math>{_\epsilon}x_n</math>, the maximal one <math>{_\epsilon}x_n^2</math> and the maximal <math>x_n^2</math> to <math>x_n</math> behaves like <math>\omega</math> to <math>{\omega}^{\tilde{2}}.\square</math>
  
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=== Speedup theorem for derivatives ===
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=== Gelfond-Schneider theorem ===
  
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For <math>\mathbb{B}_{\hat{\nu}}(0) \subset  D \subseteq {}^{\omega}\mathbb{C},</math> the Taylor series<div style="text-align:center;"><math>f(z):=f(0) + \sum\limits_{m=1}^{\omega }{\widehat{m!}\,{{f}^{(m)}}(0){z^m}},</math></div><math>b_{\varepsilon n} := \hat{\varepsilon}\,\acute{n}! = 2^j, j, n \in {}^{\omega}\mathbb{N}^{*}, \varepsilon \in ]0, r^n[, {{d}_{\varepsilon k n}}:={{\varepsilon}^{{\hat{n}}}}{e}^{\hat{n}k\tau i}</math> and <math>f</math>'s radius of convergence <math>r \in {}^{\nu}{\mathbb{R}}_{&gt;0}</math> imply<div style="text-align:center;"><math>{{f}^{(n)}}(0)=b_{\varepsilon n}\sum\limits_{k=1}^{n}{\delta_n^* f({{d}_{\varepsilon k n}})}.</math></div>
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It holds <math>a^b \notin {}_{\omega}^{\omega} \mathbb{A}_\mathbb{C}</math> where <math>a, c \in {}^{\omega} \mathbb{A}_\mathbb{C} \setminus \mathbb{B}</math> and infinitesimal <math>\varepsilon, b \in {}^{\omega}\mathbb{A}_\mathbb{C} \setminus {}_{\omega}^{\omega}\mathbb{R}</math>.
  
 
==== Proof: ====
 
==== Proof: ====
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Taylor's theorem<ref name="Remmert">[[w:Reinhold Remmert|<span class="wikipedia">Remmert, Reinhold</span>]]: ''Funktionentheorie 1'' : 3., verb. Aufl.; 1992; Springer; Berlin; ISBN 9783540552338, S. 165 f.</ref> and the properties of the roots of unity.<math>\square</math>
 
  
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== Reference ==
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The minimal polynomials <math>p</math> (and <math>q</math>) of <math>c^r</math> resp. <math>c^{r\pm\varepsilon} = a^b</math> for maximal <math>r \in {}_{\omega}^{\omega}\mathbb{R}_{>0}</math> and <math>f = p\;(q)</math> lead to the contradiction <math>{}^1f(c^{r(\pm\varepsilon)}) \ne 0 = (f(c^r) - f(c^{r\pm\varepsilon})) / (c^r - c^{r\pm\varepsilon}) = {}^1f(c^{r(\pm\varepsilon)}).\square</math>
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<references />
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== MWiki has moved ==
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The new URL is: [https://en.hwiki.de/maths.html HWiki]
  
 
== Recommended reading ==
 
== Recommended reading ==

Latest revision as of 17:01, 31 July 2024

Welcome to MWiki

Theorems of the month

Prime number theorem

For [math]\displaystyle{ \pi(x) := |\{p \in {\mathbb{P}_{\le x}} : x \in {}^{\omega}{\mathbb{R}}\}| }[/math], it holds that [math]\displaystyle{ \pi(\omega) = \widetilde{{_e}\omega}\omega + \mathcal{O}({_\epsilon}\omega\;{\omega}^{\tilde{2}}) }[/math].

Proof:

From intervals of fix length [math]\displaystyle{ y \in {}^{\omega}{\mathbb{R}_{\gt 0}}, \check{y} }[/math] 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. The Stirling formula suggests the prime gap [math]\displaystyle{ n = {\epsilon}^{\sigma} = \mathcal{O}({_\epsilon}(n!)) }[/math].

For induction basis [math]\displaystyle{ n = 2 }[/math] resp. 3, the hypothesis states the first interval to contain [math]\displaystyle{ x_n/{_\epsilon}x_n }[/math] primes for [math]\displaystyle{ n \in {}^{\omega}{\mathbb{N}_{\ge2}} }[/math] and [math]\displaystyle{ x_4 \in [2, 4[ }[/math]. Stepping from [math]\displaystyle{ x_n }[/math] to [math]\displaystyle{ x_n^2 }[/math] finds [math]\displaystyle{ \pi(x_n^2) = \pi(x_n) \check{x}_n }[/math] primes only from [math]\displaystyle{ \pi(x_n) = x_n/{_\epsilon}x_n }[/math]. The average prime gap is [math]\displaystyle{ {_\epsilon}x_n }[/math], the maximal one [math]\displaystyle{ {_\epsilon}x_n^2 }[/math] and the maximal [math]\displaystyle{ x_n^2 }[/math] to [math]\displaystyle{ x_n }[/math] behaves like [math]\displaystyle{ \omega }[/math] to [math]\displaystyle{ {\omega}^{\tilde{2}}.\square }[/math]

Gelfond-Schneider theorem

It holds [math]\displaystyle{ a^b \notin {}_{\omega}^{\omega} \mathbb{A}_\mathbb{C} }[/math] where [math]\displaystyle{ a, c \in {}^{\omega} \mathbb{A}_\mathbb{C} \setminus \mathbb{B} }[/math] and infinitesimal [math]\displaystyle{ \varepsilon, b \in {}^{\omega}\mathbb{A}_\mathbb{C} \setminus {}_{\omega}^{\omega}\mathbb{R} }[/math].

Proof:

The minimal polynomials [math]\displaystyle{ p }[/math] (and [math]\displaystyle{ q }[/math]) of [math]\displaystyle{ c^r }[/math] resp. [math]\displaystyle{ c^{r\pm\varepsilon} = a^b }[/math] for maximal [math]\displaystyle{ r \in {}_{\omega}^{\omega}\mathbb{R}_{\gt 0} }[/math] and [math]\displaystyle{ f = p\;(q) }[/math] lead to the contradiction [math]\displaystyle{ {}^1f(c^{r(\pm\varepsilon)}) \ne 0 = (f(c^r) - f(c^{r\pm\varepsilon})) / (c^r - c^{r\pm\varepsilon}) = {}^1f(c^{r(\pm\varepsilon)}).\square }[/math]

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Recommended reading

Nonstandard Mathematics

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