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Formulas (m) to (p) above lead to the following formulas for the Riemann zeta function $\zeta(s)$ and Dirichlet eta function $\eta(s)$. Formula (t) for $\eta(s)$ is much more useful than formula (s) for $\zeta(s)$ since it converges over a much wider range. Formula (t) for $\eta(s)$ can also be used to derive formulas for $\zeta(s)$ and $\eta(s)$ which converge for $\Re(s)>-1$.

$$\zeta(s)=\underset{f\to\infty}{\text{lim}}\left((2 \pi )^{s-1} \sin \left(\frac{\pi s}{2}\right)\ \Gamma (1-s) \left(-\frac{2 f^s}{s}+\sum _{n=1}^f \left(n^{s-1}+(n-1)^{s-1}\right)\right)\right),\ 1<\Re(s)<2\tag{s}$$

Formulas (m) to (p) above lead to the following formulas for the Riemann zeta function $\zeta(s)$ and Dirichlet eta function $\eta(s)$. Formula (s) and (t) for $\zeta(s)$ and $\eta(s)$ can also be used to derive formulas for $\zeta(s)$ and $\eta(s)$ which converge for $\Re(s)>-1$.

$$\zeta(s)=\underset{f\to\infty}{\text{lim}}\left(2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \left(-\frac{f^s}{s}+\frac{1}{2} \left(1+\sum\limits_{n=2}^f \left(n^{s-1}+(n-1)^{s-1}\right)\right)\right)\right),\ \Re(s)<2\tag{s}$$

$$\tilde{f}_a(x)=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f \sum\limits_{k=0}^K \frac{(-1)^k\ x^{2 k+1}}{2 k+1} \sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{f}$$

$$\tilde{f}_a'(x)=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f\sum\limits_{k=0}^K (-1)^k\ x^{2 k}\sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{g}$$

$$G_a(s)=s\int\limits_1^\infty\tilde{f}(x)\,x^{-s-1}\,dx=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f \sum\limits_{k=0}^K \frac{(-1)^k}{(2 k+1)}\frac{s}{(s-2 k-1)}\sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{j}$$

Figure (8): Illustration of $f_a(x)=\Pi(x)$ (Riemann prime counting function) where $F_a(s)=\log\zeta(s)$

$$\tilde{f}_a(x)=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f \sum\limits_{k=1}^K \frac{(-1)^k\ x^{2 k+1}}{2 k+1} \sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{f}$$

$$\tilde{f}_a'(x)=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f\sum\limits_{k=1}^K (-1)^k\ x^{2 k}\sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{g}$$

$$G_a(s)=s\int\limits_1^\infty\tilde{f}(x)\,x^{-s-1}\,dx=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f \sum\limits_{k=1}^K \frac{(-1)^k}{(2 k+1)}\frac{s}{(s-2 k-1)}\sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{j}$$

Figure (8): Illustration of $f_a(x)=\Pi(x)$ (Riemann prime counting function) where $F_a(s)=\log\zeta(s)$

For the cases $a(n)=1$, $a(n)=(-1)^{n-1}$ and $a(n)=\delta_{n-1}$ where $F_a(s)=\zeta(s)$, $F_a(s)=\eta(s)$, and $F_a(s)=1$, I've determined formulas (f) and (g) above are simply the power series for the functions defined in formulas (m) to (r) below. This proves the validity of formulas (f) and (g) above for these particular cases, but I believe formulas (f) and (g) are more generally applicable to any definition of $a(n)$ for which the Dirichlet series $F_a(s)=\sum\limits_n\frac{a(n)}{n^s}$ converges for $\Re(s)\ge 2$. All formulas below are for $x\ge 0$, but $\tilde{f}_a'(x)$ and $\tilde{f}_a(x)$ are actually even and odd functions respectively.

$\quad a(n)=1 \text{ where } F_a(s)=\zeta(s)$:

$$\tilde{f}_a'(x)=\sum\limits_n\delta(x-n)=\underset{f\to\infty}{\text{lim}}\left(-\frac{\sin (2 f \pi x)}{\pi x}+\sum\limits_{n=1}^f (\cos(2 n \pi x)+\cos(2 (n-1) \pi x))\right)\tag{m}$$

$$\tilde{f}_a(x)=\sum\limits_n\theta(x-n)=\underset{f\to\infty}{\text{lim}}\left(-\frac{\text{Si}(2 f \pi x)}{\pi}+\sum\limits_{n=1}^f \left(\frac{\sin(2 n \pi x)}{2 n \pi}+x\ \text{sinc}(2 (n-1) \pi x)\right)\right)\tag{n}$$

$\quad a(n)=(-1)^{n-1} \text{ where } F_a(s)=\eta(s)$:

$$\tilde{f}_a'(x)=\sum\limits_n (-1)^{n-1}\delta(x-n)=\underset{f\to\infty}{\text{lim}}\left(\frac{\sin(2 f \pi x)}{\pi x}-2 \sum\limits_{n=1}^f \cos((2 n-1) \pi x)\right)\tag{o}$$

$$\tilde{f}_a(x)=\sum\limits_n (-1)^{n-1}\theta(x-n)=\underset{f\to\infty}{\text{lim}}\left(\frac{\text{Si}(2 f \pi x)}{\pi }-\frac{2}{\pi}\sum\limits _{n=1}^f \frac{\sin ((2 n-1) \pi x)}{2 n-1}\right)\tag{p}$$

$\quad a(n)=\delta_{n-1} \text{ where } F_a(s)=1$:

$$\tilde{f}_a'(x)=\delta(x-1)=\underset{f\to\infty}{\text{lim}}\left(\frac{\sin(2 f \pi (x+1))}{\pi (x+1)}+\frac{\sin(2 f \pi (x-1))}{\pi (x-1)}\right)\tag{q}$$

$$\tilde{f}_a(x)=\theta(x-1)=\underset{f\to\infty}{\text{lim}}\left(\frac{\text{Si}(2 f \pi (x+1))+\text{Si}(2 f \pi (x-1))}{\pi }\right)\tag{r}$$

Formulas (m) to (p) above lead to the following formulas for the Riemann zeta function $\zeta(s)$ and Dirichlet eta function $\eta(s)$. Formula (t) for $\eta(s)$ is much more useful than formula (s) for $\zeta(s)$ since it converges over a much wider range. Formula (t) for $\eta(s)$ can also be used to derive formulas for $\zeta(s)$ and $\eta(s)$ which converge for $\Re(s)>-1$.

$$\zeta(s)=\underset{f\to\infty}{\text{lim}}\left((2 \pi )^{s-1} \sin \left(\frac{\pi s}{2}\right)\ \Gamma (1-s) \left(-\frac{2 f^s}{s}+\sum _{n=1}^f \left(n^{s-1}+(n-1)^{s-1}\right)\right)\right),\ 1<\Re(s)<2\tag{s}$$

$$\eta(s)=\underset{f\to\infty}{\text{lim}}\left(2 \pi^{s-1} \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s) \left(\frac{2^{s-1} f^s}{s}-\sum\limits_{n=1}^f (2 n-1)^{s-1}\right)\right),\ \Re(s)<2\tag{t}$$

Formulas (n), (p), (r), and (t) above are illustrated in this answer I recently posted to my own question on MathOverflow.

$$\tilde{f}_a(x)=\underset{K,f\to\infty}{\text{lim}}\left(-4 f \sum\limits_{k=0}^K \frac{(-1)^k\ x^{2 k+1}}{2 k+1} \sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{f}$$

$$\tilde{f}_a'(x)=\underset{K,f\to\infty}{\text{lim}}\left(-4 f\sum\limits_{k=0}^K (-1)^k\ x^{2 k}\sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{g}$$

I've tested formulas (f) and (g) above for the function $a(n)=\delta_{x-1}$ where conditions (1) and (2) are met, and also for the function $a(n)=\mu(n)$ where condition (1) is met and I suspect condition (2) is also met. Formulas (f) and (g) seem to evaluate correctly for both of these cases. For $a(n)=\delta_{x-1}$, the functions $\tilde{f}(x)$ and $\tilde{f}'(x)$ correspond to analytic representations of $-1+\theta(x+1)+\theta(x-1)$ and $\delta(x+1)+\delta(x-1)$. For the case $a(n)=\mu(n)$, the functions $\tilde{f}(x)$ and $\tilde{f}'(x)$ correspond to analytic representations of the Mertens function $M(x)$ and it's first-order derivative.

$$G_a(s)=s\int\limits_1^\infty\tilde{f}(x)\,x^{-s-1}\,dx=\underset{K,f\to\infty}{\text{lim}}\left(-4 f \sum\limits_{k=0}^K \frac{(-1)^k}{(2 k+1)}\frac{s}{(s-2 k-1)}\sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{j}$$

$$\tilde{f}_a(x)=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f \sum\limits_{k=0}^K \frac{(-1)^k\ x^{2 k+1}}{2 k+1} \sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{f}$$

$$\tilde{f}_a'(x)=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f\sum\limits_{k=0}^K (-1)^k\ x^{2 k}\sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{g}$$

I've tested formulas (f) and (g) above for the function $a(n)=\delta_{n-1}$ where conditions (1) and (2) are met, and also for the function $a(n)=\mu(n)$ where condition (1) is met and I suspect condition (2) is also met. Formulas (f) and (g) seem to evaluate correctly for both of these cases. For $a(n)=\delta_{n-1}$, the functions $\tilde{f}(x)$ and $\tilde{f}'(x)$ correspond to analytic representations of $-1+\theta(x+1)+\theta(x-1)$ and $\delta(x+1)+\delta(x-1)$. For the case $a(n)=\mu(n)$, the functions $\tilde{f}(x)$ and $\tilde{f}'(x)$ correspond to analytic representations of the Mertens function $M(x)$ and it's first-order derivative.

$$G_a(s)=s\int\limits_1^\infty\tilde{f}(x)\,x^{-s-1}\,dx=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f \sum\limits_{k=0}^K \frac{(-1)^k}{(2 k+1)}\frac{s}{(s-2 k-1)}\sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{j}$$