Difference between revisions of "Prime zeta P"

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The prime zeta function is defined by
 
The prime zeta function is defined by
 
$$P(z) = \displaystyle\sum_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{p^z},$$
 
$$P(z) = \displaystyle\sum_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{p^z},$$
 
where $\mathrm{Re}(z)>1$. It can be extended outside of this domain via [[analytic continuation]].
 
where $\mathrm{Re}(z)>1$. It can be extended outside of this domain via [[analytic continuation]].
  
[[File:Primezeta.png|500px]]
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<div align="center">
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<gallery>
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File:Primezetaplot.png|Graph of $P(x)$ for $x>1$.
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</gallery>
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</div>
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
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[[Derivative of prime zeta]]<br />
<strong>Theorem:</strong> The following formula holds:
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[[Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta]]<br />
$$P(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)}{k} \log \zeta(kz),$$
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where $\mu$ denotes the [[Möbius]] function, $\log$ denotes the [[logarithm]], and $\zeta$ denotes the [[Riemann zeta function]].
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=Videos=
<div class="mw-collapsible-content">
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[https://www.youtube.com/watch?v=3eN9tQX3JJ4 Zeta Function - Part 5 - Prime Zeta Function] (15 March 2012)<br />
<strong>Proof:</strong>
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</div>
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=External links=
</div>
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[http://math.stackexchange.com/questions/49383/how-does-sum-px-p-s-grow-asymptotically-for-textres-1/ How does ∑p<xp−s grow asymptotically for Re(s)<1?] <br />
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[http://math.stackexchange.com/questions/504445/zeta-question-prime-zeta-basic-calculus Zeta question - prime zeta. Basic calculus]<br />
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[http://math.stackexchange.com/questions/799590/prime-zeta-function Prime Zeta Function]<br />
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[http://math.stackexchange.com/questions/32974/prime-zeta-definition-multiplication-by-zero Prime zeta definition, multiplication by zero]<br />
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[http://math.stackexchange.com/questions/1029976/closed-form-of-prime-zeta-values Closed-form of prime zeta values]<br />
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[http://math.stackexchange.com/questions/1537551/zeros-of-the-prime-zeta-function Zeros of the prime zeta function]<br />
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[http://math.stackexchange.com/questions/246770/infinite-sum-of-powers-of-the-prime-zeta-function Infinite sum of powers of the prime zeta function]<br />
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[http://math.stackexchange.com/questions/1615626/convergence-of-prime-zeta-function-for-mathfrak-rs-1 Convergence of prime zeta function for R(s)=1?]<br />
  
 
=References=
 
=References=
Fröberg, Carl-Erik . On the prime zeta function. Nordisk Tidskr. Informationsbehandling (BIT)  8  1968 187--202.
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* {{PaperReference|The Sums of the Series of the Reciprocals of the Prime Numbers and of Their Powers|1881|Charles Watkins Merrifield}}
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* {{PaperReference|On the Sums of the Inverse Powers of the Prime Numbers|1891|James Whitbread Lee Glaisher}}
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* {{PaperReference|On the prime zeta function|1968|Carl-Erik Fröberg}}
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[[Category:SpecialFunction]]

Latest revision as of 23:29, 17 March 2017

The prime zeta function is defined by $$P(z) = \displaystyle\sum_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{p^z},$$ where $\mathrm{Re}(z)>1$. It can be extended outside of this domain via analytic continuation.


Properties

Derivative of prime zeta
Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta

Videos

Zeta Function - Part 5 - Prime Zeta Function (15 March 2012)

External links

How does ∑p<xp−s grow asymptotically for Re(s)<1?
Zeta question - prime zeta. Basic calculus
Prime Zeta Function
Prime zeta definition, multiplication by zero
Closed-form of prime zeta values
Zeros of the prime zeta function
Infinite sum of powers of the prime zeta function
Convergence of prime zeta function for R(s)=1?

References