Difference between revisions of "Riemann zeta"

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Consider the function $\zeta$ defined by the series
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The Riemann zeta function $\zeta$ is defined for $\mathrm{Re}(z)>1$ by
 
$$\zeta(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z}.$$
 
$$\zeta(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z}.$$
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Riemannzeta.png|Graph of $\zeta$ on $[-5,5]$.
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File:Riemannzetaplot.png|Graph of $\zeta$ on $[-21,10] \setminus \{1\}$.
File:Complex zeta.jpg|[[Domain coloring]] of [[analytic continuation]] of $\zeta$.
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File:Complexriemannzeta.png|[[Domain coloring]] of $\zeta$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
 
==Properties==
 
==Properties==
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[[Derivative of Riemann zeta]]<br />
<strong>Proposition:</strong> If $\mathrm{Re} \hspace{2pt} z > 1$, then the series defining $\zeta(z)$ converges.
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[[Euler product for Riemann zeta]]<br />
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[[Laurent series of the Riemann zeta function]]<br />
<strong>Proof:</strong>
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[[Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta]]<br />
</div>
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[[Series for log(riemann zeta) over primes]]<br />
</div>
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[[Series for log(Riemann zeta) in terms of Mangoldt function]]<br />
 
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[[Logarithmic derivative of Riemann zeta in terms of series over primes]]<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[[Logarithmic derivative of Riemann zeta in terms of Mangoldt function]]<br />
<strong>Proposition (Euler Product):</strong> $\zeta(z)=\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z} = \displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}}$
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[[Reciprocal Riemann zeta in terms of Mobius]]<br />
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[[Riemann zeta as integral of monomial divided by an exponential]]<br />
<strong>Proof:</strong>
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[[Riemann zeta as contour integral]]<br />
</div>
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[[Riemann zeta at even integers]]<br />
</div>
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[[Functional equation for Riemann zeta]]<br />
 
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[[Functional equation for Riemann zeta with cosine]]<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Proposition:</strong> Let $n$ be a positive integer. Then
 
$$\zeta(2n)=(-1)^{n+1}\dfrac{B_{2n}(2\pi)^{2n}}{2(2n)!},$$
 
where $B_n$ denotes the [[Bernoulli numbers]].
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong>
 
</div>
 
</div>
 
 
 
{{:Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta}}
 
  
 
=Videos=
 
=Videos=
[https://www.youtube.com/watch?v=ZlYfEqdlhk0&list=PL32446FDD4DA932C9 Riemann Zeta function playlist]<br />
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[https://www.youtube.com/watch?v=2TE6B10LmCQ The Basel Problem and $\zeta(2k)$ (11 May 2017)]<br />
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[https://www.youtube.com/watch?v=sD0NjbwqlYw Visualizing the Riemann zeta function and analytic continuation (9 December 2016)]<br />
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[https://www.youtube.com/watch?v=cFWMht03XME Zeta Integral (5 July 2016)]<br />
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[https://www.youtube.com/watch?v=Vsib1v5vfkc Möbius Inversion of $\zeta(s)$ (3 July 2016)]<br />
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[https://www.youtube.com/watch?v=ZlYfEqdlhk0&list=PL32446FDD4DA932C9 Riemann Zeta function playlist (8 March 2012)]<br />
  
 
=External links=
 
=External links=
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*[http://www.claymath.org/sites/default/files/ezeta.pdf English translation of Riemann's paper "On the number of prime numbers less than a given quantity"]
 
*[http://www.claymath.org/sites/default/files/ezeta.pdf English translation of Riemann's paper "On the number of prime numbers less than a given quantity"]
 
*[http://www.uam.es/personal_pdi/ciencias/cillerue/Curso/zeta2.pdf Evaluating $\zeta(2)$]
 
*[http://www.uam.es/personal_pdi/ciencias/cillerue/Curso/zeta2.pdf Evaluating $\zeta(2)$]
*[https://www.youtube.com/watch?v=yhtcJPI6AtY The Riemann Hypothesis: How to make $1 Million Without Getting Out of Bed]
 
 
*[http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/riemannhyp.htm The Riemann Hypothesis: FAQ and resources]
 
*[http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/riemannhyp.htm The Riemann Hypothesis: FAQ and resources]
 
*[http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/devlin.pdf How Euler discovered the zeta function]
 
*[http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/devlin.pdf How Euler discovered the zeta function]
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*[http://www.dtc.umn.edu/~odlyzko/zeta_tables/ Andrew Odlyzko: Tables of zeros of the Riemann zeta function]
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=See also=
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[[Reciprocal Riemann zeta]]
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=References=
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* {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|next=Euler product for Riemann zeta}}: § Introduction $(1)$
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* {{BookReference|Higher Transcendental Functions Volume III|1953|Harry Bateman|prev=findme|next=Euler product for Riemann zeta}}: pg. $170$
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Euler product for Riemann zeta}}: $23.2.1$
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{{:Number theory functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 18:40, 12 May 2017

The Riemann zeta function $\zeta$ is defined for $\mathrm{Re}(z)>1$ by $$\zeta(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z}.$$

Properties

Derivative of Riemann zeta
Euler product for Riemann zeta
Laurent series of the Riemann zeta function
Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta
Series for log(riemann zeta) over primes
Series for log(Riemann zeta) in terms of Mangoldt function
Logarithmic derivative of Riemann zeta in terms of series over primes
Logarithmic derivative of Riemann zeta in terms of Mangoldt function
Reciprocal Riemann zeta in terms of Mobius
Riemann zeta as integral of monomial divided by an exponential
Riemann zeta as contour integral
Riemann zeta at even integers
Functional equation for Riemann zeta
Functional equation for Riemann zeta with cosine

Videos

The Basel Problem and $\zeta(2k)$ (11 May 2017)
Visualizing the Riemann zeta function and analytic continuation (9 December 2016)
Zeta Integral (5 July 2016)
Möbius Inversion of $\zeta(s)$ (3 July 2016)
Riemann Zeta function playlist (8 March 2012)

External links

See also

Reciprocal Riemann zeta

References

Number theory functions