Question (1): Is there a closed form representation for $g(x)$ defined in formula (1) below?
$$g(x)=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N \frac{(-1)^{n+1}\, (2 n+1)}{n!}\, \pi^n\, \frac{\zeta'(2 n+1)}{\zeta(2 n+1)} x^{-2 n}\right)\tag{1}$$
Figure (1) below illustrates formula (1) for $g(x)$ above evaluated at $N=100$. I suspect $g(x)$ defined in formula (1) converges for $x>0$ as $N\to\infty$.
Figure (1): Illustration of formula (1) for $g(x)$
Question (2): Is anything known about the rationality of $\frac{\zeta'(2 n+1)}{\zeta(2 n+1)}$ or $\pi^n\, \frac{\zeta'(2 n+1)}{\zeta(2 n+1)}$?
The context of this question is an investigation related to
$$p(x)=\underset{K\to\infty}{\text{lim}}\left(\sum\limits_{k=1}^K \Lambda(k) \left(\frac{2 \pi k^2}{x^2}-1\right) e^{-\frac{\pi k^2}{x^2}}\right)\tag{2}$$
where
$$P(s)=s\, \mathcal{M}_x[p(x)](-s)=s \int\limits_0^\infty p(x)\, x^{-s-1}\, ds$$ $$=\pi^{-\frac{s}{2}}\, (1-s)\, \Gamma\left(\frac{s}{2}+1\right)\,\frac{\zeta'(s)}{\zeta(s)}\,,\quad\Re(s)>1\tag{3}$$
The residue formula for $p(x)$, which I believe converges for $x>0$, is
$$p_o(x)=\log(2 \pi)+p_{-2n}(x)+p_\rho(x)\tag{4}$$
where
$$p_{-2n}(x)=\sum\limits_{n=1}^{\infty} \frac{x^{-2 n} (-\pi)^n \left(\zeta'(-2 n)+n (2 n+1) \left(\zeta''(-2 n)+\zeta'(-2 n) (\psi ^{(0)}(n)+2 \log(x)-\log(\pi))\right)\right)}{2 n n! \zeta'(-2 n)}\tag{5}$$
is a sum over the trivial zeta-zeros and
$p_\rho(x)=\sum\limits_{\rho} \left(\pi^{-\frac{\rho}{2}} \left(1-\rho\right) \, \Gamma\left(\frac{\rho}{2}+1\right) \frac{x^{\rho}}{\rho}\right)\tag{6}$
is a sum over the non-trivial zeta-zeros.
I'm investigating a potential pseudo-closed-form (meaning non-elementary functions are acceptable) for the function $p_{-2n}(x)$ defined in formula (5) above which I believe can also be evaluated as illustrated in formula (7) below
$$p_{-2n}(x)=r(x)+f(x)\tag{7}$$
where
$$r(x)=-\frac{1}{2 x^4} \left(3 \pi x^2 \text{1F1}^{(1,0,0)}\left(1,2,-\frac{\pi }{x^2}\right)-\pi ^2 \text{1F1}^{(1,0,0)}\left(2,3,-\frac{\pi }{x^2}\right)+x^4 \left(E_1\left(\frac{\pi }{x^2}\right)+\log \left(\frac{1}{x^2}\right)+2 \log (x)-2\right)+e^{-\frac{\pi }{x^2}} x^2 \left(x^2 (\gamma +2+\log (\pi ))+\left(4 \pi -2 x^2\right) \log (x)-2 \pi (\gamma -1+\log (\pi ))\right)\right)\tag{8}$$
and
$$f(x)=\sum\limits_{n=1}^{\infty} \frac{(-1)^n\, (2 n+1)}{2 n!}\, \pi^n\, \frac{\zeta''(-2 n)}{\zeta'(-2 n)}\, x^{-2 n}\tag{9}.$$
My initial question was focused on a closed form for $f(x)$ defined in formula (9) above, but after further investigation it seemed to me the series representation of $f(x)$ defined in formula (9) above can be split into
$$f(x)=g(x)+h(x)\tag{9a}$$
where $g(x)$ is defined in formula (1) above (i.e. the subject of this question) and
$$h(x)=\sum\limits_{n=1}^\infty \frac{(-1)^n\, (2 n+1)}{n!}\, \pi^n \left(\gamma+\log(2 \pi)-H_{2 n}\right)\, x^{-2 n}\tag{10}$$
which I believe can be evaluated as
$$h(x)=\frac{1}{2 x^2} e^{-\frac{\pi }{x^2}} \left(x^2 \, _1F_1^{(1,0,0)}\left(0,\frac{1}{2},\frac{\pi }{x^2}\right)-2 \pi \, _1F_1^{(1,0,0)}\left(0,\frac{3}{2},\frac{\pi }{x^2}\right)+e^{\frac{\pi }{x^2}} \left(2 \pi \, _1F_1^{(1,0,0)}\left(2,2,-\frac{\pi }{x^2}\right)-x^2 \, _1F_1^{(1,0,0)}\left(1,1,-\frac{\pi }{x^2}\right)\right)-2 \left(e^{\frac{\pi }{x^2}}-1\right) x^2 (\gamma +\log (2 \pi ))+\pi (-4 \gamma +6-4 \log (2 \pi ))\right)\tag{11}.$$
The appearance of $\frac{\zeta'(2 n+1)}{\zeta(2 n+1)}$ in formula (1) above is perhaps related to the appearance of $\frac{\zeta'(s)}{\zeta(s)}$ in formula (3) above.
In summary, I'm currently working with the formula
$$p_{-2n}(x)=r(x)+h(x)+g(x)\tag{12}$$
where I use the series representation in formula (1) above to evaluate $g(x)$ and the pseudo-closed-forms in formulas (8) and (11) above to evaluate $r(x)$ and $h(x)$.
I've been struggling to define an accurate and simple asymptotic for $p(x)$ defined in formula (2) above, but the only accurate asymptotic I've come up with is $\log(2 \pi)+p_{-2n}(x)$ where formula (12) above is used to evaluate $p_{-2n}(x)$, and this is not yet a closed-form much less simple.
Figure (2) below illustrates formula (2) above for $p(x)$ (blue curve) seems to converge to $\log(2 \pi)$ (orange horizontal dashed line) as $x\to\infty$ which is consistent with the leading term in the explicit formula for $p_o(x)$ defined in formula (4) above.
Figure (2): Illustration of formula (2) for $p(x)$
Figure (3) below illustrates formula (6) above for $p_\rho(x)$ evaluated over the first $100$ pairs of non-trivial zeta-zeros (blue curve) which is largely dominated by the contribution of the first pair of non-trivial zeta-zeros.
Figure (3): Illustration of formula (6) for $p_\rho(x)$