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==== Proof: ==== | ==== Proof: ==== | ||
− | 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 | + | 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> |
== Recommended reading == | == Recommended reading == |
Revision as of 01:51, 14 June 2020
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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\[\mathbb{A}(m, n) = \widehat{\zeta(\grave{m})}\,z(m){{(2n+1)}^{m}}\left( n+\mathcal{O}({_e}n) \right),\]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
References
- ↑ Scheid, Harald: Zahlentheorie : 1st Ed.; 1991; Bibliographisches Institut; Mannheim; ISBN 9783411148417, p. 323.