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November 5th, 2016, 04:41 PM  #1 
Newbie Joined: Sep 2016 From: Tempe, Arizona Posts: 6 Thanks: 1  What is Relationship between $\zeta(s)$ and Simple PrimePower Counting Function?
Assume the following definitions: $\zeta(s)$  Riemann zeta function $\zeta'(s)$  firstorder derivative of the Riemann zeta function $\vartheta(s)$  first Chebyshev function $\vartheta'(s)$  firstorder derivative of the first Chebyshev function $\psi(s)$  second Chebyshev function $\psi'(s)$  firstorder derivative of the second Chebyshev function $\pi(x)$  fundamental prime counting function which takes a unit step at each prime $\pi'(x)$  firstorder derivative of the fundamental prime counting function $J(x)$  Riemann's primepower counting function which takes a step of $1/n$ at each primepower $x=p^n$ $J'(x)$  firstorder derivative of Riemann's primepower counting function $K(x)$  simple primepower counting function which takes a unit step at each primepower $K'(x)$  firstorder derivative of the simple primepower counting function Riemann defined the following relationships between $\pi(x)$ and $J(x)$: $$J(x)=\sum_{n=1}^{Floor(Log(2,x))}\frac{1}{n}\pi( x^{\frac{1}{n}})$$ $$\pi(x)=\sum_{n=1}^{Floor(Log(2,x))}\frac{\mu(n)} {n}J(x^{\frac{1}{n}})$$ The relationship between $\pi(x)$ and $K(x)$ is slightly simpler (the $\frac{1}{n}$ term is removed): $$K(x)=\sum_{n=1}^{Floor(Log(2,x))}\pi(x^{\frac{1} {n}})$$ $$\pi(x)=\sum_{n=1}^{Floor(Log(2,x))}\mu(n)K(x^{\f rac{1}{n}})$$ Note the relationship between $\pi(x)$ and $K(x)$ is analogous to the relationship between $\vartheta(x)$ and $\psi(x)$: $$\psi(x)=\sum_{n=1}^{Floor(Log(2,x))}\vartheta(x^ {\frac{1}{n}})$$ $$\vartheta(x)=\sum_{n=1}^{Floor(Log(2,x))}\mu(n)\ psi(x^{\frac{1}{n}})$$ The relationship between $J(x)$ and $K(x)$ is a bit more complex: $$K(x)=\sum_{n=1}^{Floor(Log(2,x))}\frac{\phi(gsfd (n))}{gsfd(n)}J(x^{\frac{1}{n}})$$ $$J(x)=\sum_{n=1}^{Floor(Log(2,x))}\frac{\mu(gsfd( n))\phi(n)gsfd(n)}{n^2}K(x^{\frac{1}{n}})$$ The $\phi$ function above is Euler's totient function and the $gsfd(n)$ function above is the greatest squarefree divisor of $n$ (which is also known as the radical or squarefree kernel of $n$). There are relationships between $\zeta(s)$ and $\pi(x)$, $J(x)$, $J'(x)$, $\psi(x)$, and $\psi'(x)$ such as the following: $$\log\zeta(s)=s\int_{0}^{\infty}\frac{\pi(x)}{x(x ^s1)}dx$$ $$\log\zeta(s)=s\int_{0}^{\infty}J(x)x^{s1}dx$$ $$\log\zeta(s)=\int_{0}^{\infty}x^{s}J'(x)dx$$ $$\frac{\zeta(s)}{(\zeta(s)}=s\int_{0}^{\infty}\psi( x)x^{s1}dx$$ $$\frac{\zeta'(s)}{\zeta(s)}=\int_{0}^{\infty}x^{s}\psi'(x)dx$$ My questions are: 1) What is the relationship between $\zeta(s)$ and the simple primepower counting function $K(x)$? 2) What is the relationship between $\zeta(s)$ and the first order derivative $K'(x)$ of the simple primepower counting function? I'm looking for integral relationships such as those specified above for $\pi(x)$, $J(x)$, $J'(x)$, $\psi(x)$, and $\psi'(x)$. 

Tags 
$zetas$, counting, function, mellin, numbertheory, prime numbers, primepower, relationship, riemannzeta, simple 
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