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November 5th, 2016, 04:41 PM   #1
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What is Relationship between $\zeta(s)$ and Simple Prime-Power Counting Function?

Assume the following definitions:
$\zeta(s)$ - Riemann zeta function
$\zeta'(s)$ - first-order derivative of the Riemann zeta function

$\vartheta(s)$ - first Chebyshev function
$\vartheta'(s)$ - first-order derivative of the first Chebyshev function

$\psi(s)$ - second Chebyshev function
$\psi'(s)$ - first-order derivative of the second Chebyshev function

$\pi(x)$ - fundamental prime counting function which takes a unit step at each prime
$\pi'(x)$ - first-order derivative of the fundamental prime counting function

$J(x)$ - Riemann's prime-power counting function which takes a step of $1/n$ at each prime-power $x=p^n$
$J'(x)$ - first-order derivative of Riemann's prime-power counting function

$K(x)$ - simple prime-power counting function which takes a unit step at each prime-power
$K'(x)$ - first-order derivative of the simple prime-power 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 square-free divisor of $n$ (which is also known as the radical or square-free 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:
$$\log\zeta(s)=s\int_{0}^{\infty}\frac{\pi(x)}{x(x ^s-1)}dx$$
$$\log\zeta(s)=s\int_{0}^{\infty}J(x)x^{-s-1}dx$$
$$\log\zeta(s)=\int_{0}^{\infty}x^{-s}J'(x)dx$$
$$-\frac{\zeta(s)}{(\zeta(s)}=s\int_{0}^{\infty}\psi( x)x^{-s-1}dx$$
$$-\frac{\zeta'(s)}{\zeta(s)}=\int_{0}^{\infty}x^{-s}\psi'(x)dx$$

My questions are:
1) What is the relationship between $\zeta(s)$ and the simple prime-power counting function $K(x)$?
2) What is the relationship between $\zeta(s)$ and the first order derivative $K'(x)$ of the simple prime-power counting function?

I'm looking for integral relationships such as those specified above for $\pi(x)$, $J(x)$, $J'(x)$, $\psi(x)$, and $\psi'(x)$.
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