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= Welcome to MWiki =
 
= Welcome to MWiki =
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== Theorems of the month ==
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== Theorem of the month ==
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First fundamental theorem of exact differential and integral calculus for line integrals: The function <math>F(z)=\int\limits_{\gamma }{f(\zeta )dB\zeta }</math> where <math>\gamma: [d, x[C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}, C \subseteq \mathbb{R}, f: A \rightarrow {}^{(\omega)}\mathbb{K}, d \in [a, b[C</math>, and choosing <math>\curvearrowright B \gamma(x) = \gamma(\curvearrowright D x)</math> is exactly <math>B</math>-differentiable, and for all <math>x \in [a, b[C</math> and <math>z = \gamma(x)</math>
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=== Counting theorem for algebraic numbers ===
  
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<div style="text-align:center;"><math>F' \curvearrowright B(z) = f(z).</math></div>
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The number <math>\mathbb{A}(m, n)</math> of algebraic numbers of polynomial or series degree <math>m</math> and thus in general for the Riemann zeta function <math>\zeta</math> asymptotically satisfies the equation <math>\mathbb{A}(m, n) = \widetilde{\zeta(\grave{m})}\,z(m){{(2n+1)}^{m}}\left( n+\mathcal{O}({_e}n) \right)</math>, where <math>z(m)</math> is the average number of zeros of a polynomial or series.
  
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==== Proof: ====
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The case <math>m = 1</math> requires by <ref name="Scheid">[[w:Harald Scheid|<span class="wikipedia">Scheid, Harald</span>]]: ''Zahlentheorie'' : 1st Ed.; 1991; Bibliographisches Institut; Mannheim; ISBN 9783411148417, p. 323.</ref> the error term <math>\mathcal{O}({_e}n n)</math> and represents the number <math>4\sum\limits_{k=1}^{n}{\varphi (k)}-1</math> by the <math>\varphi</math>-function. For <math>m > 1</math>, the divisibility conditions neither change the error term <math>\mathcal{O}({_e}n)</math> nor the leading term. Polynomials or series such that <math>\text{gcd}({a}_{0}, {a}_{1}, ..., {a}_{m}) \ne 1</math> are excluded by <math>1/\zeta(\grave{m})</math>: The latter is given by taking the product over the prime numbers <math>p</math> of all <math>(1 - {p}^{-\grave{m}})</math> absorbing here multiples of <math>p</math> and representing sums of geometric series.<math>\square</math>
  
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Proof: <math>dB(F(z))=\int\limits_{t\in [d,x]C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}-\int\limits_{t\in [d,x[C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{x}{f(\gamma (t))\frac{\gamma (\curvearrowright Dt)-\gamma (t)}{\curvearrowright Dt-t}dDt}=f(\gamma (x)){{{\gamma }'}_{\curvearrowright }}D(x)dDx=\,f(\gamma (x))(\curvearrowright B\gamma (x)-\gamma (x))=f(z)dBz.\square</math>
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== Recommended reading ==
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Second fundamental theorem of exact differential and integral calculus for line integrals: According to the conditions from above, we have with <math>\gamma: [a, b[C \rightarrow {}^{(\omega)}\mathbb{K}</math> that
 
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<div style="text-align:center;"><math>F(\gamma (b))-F(\gamma (a))=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.</math></div>
 
  
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[https://en.calameo.com/books/003777977258f7b4aa332 Nonstandard Mathematics]
  
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Proof: <math>F(\gamma (b))-F(\gamma (a))=\sum\limits_{t\in [a,b[C}{F(\curvearrowright B\,\gamma (t))}-F(\gamma (t))=\sum\limits_{t\in [a,b[C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t))(\curvearrowright B\,\gamma (t)-\gamma (t))}=\int\limits_{t\in [a,b[C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.\square</math>
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== References ==
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<references />
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== Recommended readings ==
 
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[https://en.calameo.com/books/003777977258f7b4aa332 Nonstandard Mathematics]
 
  
 
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Revision as of 23:23, 30 June 2022

Welcome to MWiki

Theorem of the month

Counting theorem for algebraic numbers

The number [math]\displaystyle{ \mathbb{A}(m, n) }[/math] of algebraic numbers of polynomial or series degree [math]\displaystyle{ m }[/math] and thus in general for the Riemann zeta function [math]\displaystyle{ \zeta }[/math] asymptotically satisfies the equation [math]\displaystyle{ \mathbb{A}(m, n) = \widetilde{\zeta(\grave{m})}\,z(m){{(2n+1)}^{m}}\left( n+\mathcal{O}({_e}n) \right) }[/math], where [math]\displaystyle{ z(m) }[/math] is the average number of zeros of a polynomial or series.

Proof:

The case [math]\displaystyle{ m = 1 }[/math] requires by [1] the error term [math]\displaystyle{ \mathcal{O}({_e}n n) }[/math] and represents the number [math]\displaystyle{ 4\sum\limits_{k=1}^{n}{\varphi (k)}-1 }[/math] by the [math]\displaystyle{ \varphi }[/math]-function. For [math]\displaystyle{ m \gt 1 }[/math], the divisibility conditions neither change the error term [math]\displaystyle{ \mathcal{O}({_e}n) }[/math] nor the leading term. Polynomials or series such that [math]\displaystyle{ \text{gcd}({a}_{0}, {a}_{1}, ..., {a}_{m}) \ne 1 }[/math] are excluded by [math]\displaystyle{ 1/\zeta(\grave{m}) }[/math]: The latter is given by taking the product over the prime numbers [math]\displaystyle{ p }[/math] of all [math]\displaystyle{ (1 - {p}^{-\grave{m}}) }[/math] absorbing here multiples of [math]\displaystyle{ p }[/math] and representing sums of geometric series.[math]\displaystyle{ \square }[/math]

Recommended reading

Nonstandard Mathematics

References

  1. ↑ Scheid, Harald: Zahlentheorie : 1st Ed.; 1991; Bibliographisches Institut; Mannheim; ISBN 9783411148417, p. 323.
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