Questions about the Dirichlet series $K(s)=\sum\limits_{p^k} p^{\,-k\,s}$

Question

This question is related to Riemann's prime-power counting function $J(x)$, the fundamental prime-counting function $\pi(x)$, and the simple prime-power counting function $k(x)$ defined below where $p\in\mathbb{P}$ and $k\in\mathbb{Z}$.

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(1) $\quad J(x)=\sum\limits_{n=1}^x a_J(n)\,,\qquad a_J(n)=\cases{\frac{1}{k},& $n=p^k$ \\ 0,& otherwise}$

(2) $\quad\pi(x)=\sum\limits_{n=1}^x a_{\pi}(n)\,,\qquad a_{\pi}(n)=\cases{1,& $n\in\mathbb{P}$ \\ 0,& otherwise}$

(3) $\quad k(x)=\sum\limits_{n=1}^x a_k(n)\,,\qquad a_k(n)=\cases{1,& $n=p^k$ \\ 0,& otherwise}$

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The three prime counting functions defined above are related as follows where $rad(n)$ is the radical of $n$ (also referred to as the square-free kernel of $n$) which is the greatest square-free divisor of $n$.

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(4) $\quad J(x)=\sum\limits_{n=1}^{\log_2(x)}\frac{1}{n}\,\pi\left(x^{\frac{1}{n}}\right)$

(5) $\quad \pi(x)=\sum\limits_{n=1}^{\log_2(x)}\frac{\mu(n)}{n}\,J\left(x^{\frac{1}{n}}\right)$

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(6) $\quad K(x)=\sum\limits_{n=1}^{\log_2(x)}\pi\left(x^\frac{1}{n}\right)$

(7) $\quad \pi(x)=\sum\limits_{n=1}^{\log_2(x)}\mu(n)\,K\left(x^{\frac{1}{n}}\right)$

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(8) $\quad K(x)=\sum\limits_{n=1}^{\log_2(x)}\frac{\phi(n)}{n}\,J\left(x^\frac{1}{n}\right)$

(9) $\quad J(x)=\sum\limits_{n=1}^{\log_2(x)}\frac{\mu(rad(n))\,\phi(rad(n))}{n}\,K\left(x^{\frac{1}{n}}\right)$

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The three Dirichlet series defined below are related to the three prime-counting functions defined in (1) to (3) above.

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(10) $\quad \log\zeta(s)=\sum\limits_{n=1}^\infty \frac{a_J(n)}{n^s}\,,\quad\Re(s)>1$

(11) $\quad P(s)=\sum\limits_{n=1}^\infty \frac{a_{\pi}(n)}{n^s}\,,\qquad\Re(s)>1\qquad\text{(prime zeta function)}$

(12) $\quad K(s)=\sum\limits_{n=1}^\infty \frac{a_k(n)}{n^s}\,,\qquad\Re(s)>1$

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I believe the $\log\zeta(s)$ function defined in (10) above can be analytically continued to the entire complex plane with branch points at $s=1$ and the zeros of $\zeta(s)$, and the prime zeta function $P(s)$ defined in (11) above can be analytically continued to the strip $0<Re(s)\le 1$ but not beyond the line $\Re(s)=0$. I've read very little is known about the zeros of the prime zeta function $P(s)$.

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Question (1): Can the $K(s)$ function defined in (12) above be analytically continued beyond the line $\Re(s)=1$ and if so, what are the limits of this continuation?

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Question (2): Is much known about the zeros of the $K(s)$ function defined in (12) above?

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The prime zeta function $P(s)$ can be analytically continued as illustrated in formula (13) below. Based on formulas (5) and (8) above and formula (13) below, my guess is the function $K(s)$ can be analytically continued as illustrated in formula (14) below.

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(13) $\quad P(s)=\sum\limits_{n=1}^\infty\frac{\mu(n)}{n}\,\log\zeta(n s)\,,\qquad\Re(s)>0$

(14) $\quad K(s)\stackrel{?}{=}\sum\limits_{n=1}^\infty\frac{\phi(n)}{n}\,\log\zeta(n s)\,,\qquad\Re(s)>0?$

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The following figure illustrates the imaginary part of formula (14) for $K(s)$ above in the interval $0<\Re(s)<1.2$ where the series in formula (14) is evaluated over the first 200 terms. Note $\Im(K(s))$ seems to take a negative step of magnitude $\frac{\phi(k)}{k}\pi$ at $s=\frac{1}{k}$ where $k\in\mathbb{Z}$.

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Figure (1): Illustration of formula (14) for $\Im(K(s))$

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Question (3): Is it true the $K(s)$ function defined in formula (12) above can be analytically continued in the strip $0<Re(s)\le 1$ as illustrated in formula (14) above?