Difference between revisions of "Prime zeta P"

From specialfunctionswiki
Jump to: navigation, search
(Created page with "The prime zeta function is defined by $$P(z) = \displaystyle\sum_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{p^z},$$ where $\mathrm{Re}(z)>1$.")
 
(Videos)
 
(26 intermediate revisions by the same user not shown)
Line 1: Line 1:
 +
__NOTOC__
 
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 $\mathrm{Re}(z)>1$.
+
where $\mathrm{Re}(z)>1$. It can be extended outside of this domain via [[analytic continuation]].
 +
 
 +
 
 +
<div align="center">
 +
<gallery>
 +
File:Primezetaplot.png|Graph of $P(x)$ for $x>1$.
 +
</gallery>
 +
</div>
 +
 
 +
=Properties=
 +
[[Derivative of prime zeta]]<br />
 +
[[Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta]]<br />
 +
 
 +
=Videos=
 +
[https://www.youtube.com/watch?v=3eN9tQX3JJ4 Zeta Function - Part 5 - Prime Zeta Function] (15 March 2012)<br />
 +
 
 +
=External links=
 +
[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 />
 +
[http://math.stackexchange.com/questions/504445/zeta-question-prime-zeta-basic-calculus Zeta question - prime zeta. Basic calculus]<br />
 +
[http://math.stackexchange.com/questions/799590/prime-zeta-function Prime Zeta Function]<br />
 +
[http://math.stackexchange.com/questions/32974/prime-zeta-definition-multiplication-by-zero Prime zeta definition, multiplication by zero]<br />
 +
[http://math.stackexchange.com/questions/1029976/closed-form-of-prime-zeta-values Closed-form of prime zeta values]<br />
 +
[http://math.stackexchange.com/questions/1537551/zeros-of-the-prime-zeta-function Zeros of the prime zeta function]<br />
 +
[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 />
 +
[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=
 +
* {{PaperReference|The Sums of the Series of the Reciprocals of the Prime Numbers and of Their Powers|1881|Charles Watkins Merrifield}}
 +
* {{PaperReference|On the Sums of the Inverse Powers of the Prime Numbers|1891|James Whitbread Lee Glaisher}}
 +
* {{PaperReference|On the prime zeta function|1968|Carl-Erik Fröberg}}
 +
 
 +
[[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