This question assumes the following definitions of prime-counting functions where $p$ denotes a prime number, $k$ denotes a positive integer, and $\theta(y)$ is the Heaviside step function which takes a unit step at $y=0$.

(1) $\quad\pi(x)=\sum\limits_{p}\theta(x-p)\,\qquad\quad\text{(fundamental prime counting function)}$

(2) $\quad\Pi(x)=\sum\limits_{n=p^k}\frac{1}{k}\,\theta(x-n)\,\quad\text{(Riemann's prime-power counting function)}$

(3) $\quad k(x)=\sum\limits_{n=p^k}\theta(x-n)\,\qquad\text{(simple prime-power counting function)}$

The following plot illustrates $\pi(x)$, $\Pi(x)$, and $k(x)$ in blue, orange, and green respectively. Note that $\pi(x)$ takes a step of $1$ at each prime and $k(x)$ takes a step of $1$ at each prime-power. $\Pi(x)$ is more complicated in that it takes a step of $\frac{1}{k}$ at each prime-power $p^k$.

Illustration of Prime Counting Functions

The $\pi(x)$ and $\Pi(x)$ functions defined above are related to $\log\zeta(s)$ as defined below.

(4) $\quad\log\zeta(s)=s\int\limits_0^\infty\Pi(x)\,x^{-s-1}\,dx\,,\quad\Re(s)>1$

(5) $\quad\log\zeta(s)=s\int\limits_0^\infty\frac{\pi(x)}{x\,\left(x^s-1\right)}\,dx\,,\qquad\quad\Re(s)>1$

Question: What is the relationship between $k(x)$ and $\log\zeta(s)$? More specifically, what is the definition of the function $f(x)$ consistent with (6) below?

(6) $\quad\log\zeta(s)=s\int\limits_0^\infty k(x)\,f(x)\,dx\,,\qquad\quad\Re(s)>1$

The following relationships between $\pi(x)$, $\Pi(x)$, and $k(x)$ may provide some insight where $rad(n)$ is the greatest square-free divisor of $n$ also referred to as the square-free kernel of $n$.

(7) $\quad\Pi(x)=\sum\limits_{n=1}^{\log_2(x)}\frac{1}{n}\,\pi(x^{1/n})$

(8) $\quad\pi(x)=\sum\limits_{n=1}^{\log_2(x)}\frac{\mu(n)}{n}\,\Pi(x^{1/n})$

(9) $\quad k(x)=\sum\limits_{n=1}^{\log_2(x)}\pi(x^{1/n})$

(10) $\quad\pi(x)=\sum\limits_{n=1}^{\log_2(x)}\mu(n)\,k(x^{1/n})$

(11) $\quad k(x)=\sum\limits_{n=1}^{\log_2(x)}\frac{\phi(n)}{n}\,\Pi(x^{1/n})$

(12) $\quad\Pi(x)=\sum\limits_{n=1}^{\log_2(x)}\frac{\mu(rad(n))\,\phi(rad(n))}{n}\,k(x^{1/n})$

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    – Aloizio Macedo
    Jan 14, 2019 at 19:07


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