The function $f_a(x)=\sum\limits_{n\le x}a(n)$ defined in formula (1) in the question above can also be evaluated as defined in formula (a) below which leads to the analytic representations of $\tilde{f}_a(x)=\underset{\epsilon\to 0}{\text{lim}}\frac{f_a(x-\epsilon)+f_a(x+\epsilon)}{2}$ and it's first order derivative $\tilde{f}_a'(x)$ defined in formulas (b) and (c) below where the evaluation frequency $f$ is assumed to be a positive integer.
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$$f(x)=\sum\limits_{n=1}^x a(n)=\sum\limits_{n=1}^x b(n)\ \left\lfloor \frac{x}{n}\right\rfloor\quad\text{where}\quad b(n)=\sum\limits_{d|n}a(d)\,\mu\left(\frac{n}{d}\right)\tag{a}$$
$$\tilde{f}_a(x)=\underset{N,f\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N b(n)\left(\frac{x}{n}-\left(\frac{1}{2}-\frac{1}{\pi}\sum\limits_{k=1}^{f\,n}\frac{\sin\left(\frac{2 \pi k x}{n}\right)}{k}\right)\right)\right),\quad x>0\tag{b}$$
$$\tilde{f}_a'(x)=\underset{N,f\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N\frac{b(n)}{n}\left(1+2\sum\limits_{k=1}^{f\,n}\cos\left(\frac{2 \pi k x}{n}\right)\right)\right),\quad x>0\tag{c}$$
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Formulas (b) and (c) above correspond to analytic representations of $f(x)=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N a(n)\,\theta(x-n)\right)$ and $f'(x)=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N a(n)\,\delta(x-n)\right)$ where $\theta(x)$ is the Heaviside step function and $\delta(x)$ is the Dirac delta function.
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When conditions (1) and (2) defined in the question above are met formulas (b) and (c) above can be simplified to formulas (d) and (e) below. The function $\tilde{f}_a'(x)$ defined in formula (e) below evaluates exactly to $2\,f\, a(n)$ when $x=n\in\mathbb{Z}\land0<|n|\le N$ which leads to the analytic formula for $\tilde{a}(x)$ defined in formula (5) in the question above. Formula (5) in the question above is simply formula (e) below multiplied by the normalization factor $\frac{1}{2\,f}$ and then evaluated at $f=1$.
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$$\tilde{f}_a(x)=\underset{N,f\to\infty}{\text{lim}}\left(\frac{1}{\pi}\sum\limits_{n=1}^N b(n)\sum\limits_{k=1}^{f\,n}\frac{\sin\left(\frac{2 \pi k x}{n}\right)}{k}\right),\quad x>0\tag{d}$$
$$\tilde{f}_a'(x)=\underset{N,f\to\infty}{\text{lim}}\left(2\sum\limits_{n=1}^N\frac{b(n)}{n}\sum\limits_{k=1}^{f\,n}\cos\left(\frac{2 \pi k x}{n}\right)\right),\quad x>0\tag{e}$$
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I've discovered the relationship between formula (4) in the question above and formula (e) above is deeper than I originally anticipated. I believe the following two analytic formulas for $\tilde{f}_a(x)$ and $\tilde{f}_a'(x)$ are valid when conditions (1) and (2) in the question above are met where once again the evaluation frequency $f$ is assumed to be a positive integer.
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$$\tilde{f}_a(x)=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f \sum\limits_{k=1}^K \frac{(-1)^k\ x^{2 k+1}}{2 k+1} \sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{f}$$
$$\tilde{f}_a'(x)=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f\sum\limits_{k=1}^K (-1)^k\ x^{2 k}\sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{g}$$
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When conditions (1) and (2) defined in the question above are met, I believe formulas (f) and (g) above are equivalent to formulas (d) and (e) above. When condition (1) is met but condition (2) is not met, I originally anticipated formulas (f) and (g) would still be valid even though formulas (d) and (e) above are no longer valid. When condition (1) is not met, I originally suspected additional terms might need to be added to formulas (f) and (g) to make them valid. But to my surprise formulas (f) and (g) above seem to evaluate correctly for every definition of $a(n)$ I've tested, but I'll note these evaluations were over very small ranges of $x$ using very small evaluation limits.
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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.
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I've tested formulas (f) and (g) above for the function $a(n)=1$ associated with the Dirichlet series for $\zeta(s)$. In this case, formula (f) above for $\tilde{f}(x)$ seems to evaluate correctly, and formula (g) above for $\tilde{f}'(x)$ seems to evaluate to a Dirac comb with a tooth missing at $x=0$. This seems to suggest the following analytic formulas for $\theta(x)$ and $\delta(x)$ where the functions $\tilde{f}_a(x)$ and $\tilde{f}_a'(x)$ referenced in formulas (h) and (i) below are defined in formulas (f) and (g) above and are assumed to be evaluated with $F_a(s)=\zeta(s)$ and with the same evaluation frequency $f$ used to evaluate formulas (h) and (i) below.
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$$\theta(x)=\underset{f\to\infty}{\text{lim}}\left(\frac{1}{2}+\frac{1}{\pi}\sum\limits_{k=1}^f \frac{\sin(2 \pi k x)}{k}-\left(\tilde{f}_a(x)-x\right)\right)\tag{h}$$
$$\delta(x)=\underset{f\to\infty}{\text{lim}}\left(2\sum\limits_{k=1}^f \cos(2 \pi k x)-\left(\tilde{f}_a'(x)-1\right)\right)\tag{i}$$
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Figures (1) and (2) below illustrate formulas (h) and (i) for $\theta(x)$ and $\delta(x)$ above both seem to evaluate pretty much as expected where both formulas were evaluated with $f=4$ and $K=200$.
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[![Illustration of Formula (h) for theta(x)][1]][1]
**Figure (1)**: Illustration of Formula (h) for $\theta(x)$ evaluated at $f=4$ and $K=200$
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[![Illustration of Formula (i) for delta(x)][2]][2]
**Figure (2)**: Illustration of Formula (i) for $\delta(x)$ evaluated at $f=4$ and $K=200$
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I've also tested formulas (f) and (g) above for the function $a(n)=(-1)^{n-1}$ associated with the Dirichlet series for $\eta(s)$. In this case, formula (f) above for $\tilde{f}(x)$ seems to evaluate correctly, and formula (g) above for $\tilde{f}'(x)$ seems to evaluate to the sum of two Dirac combs where one of them has a tooth missing at $x=0$.
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I've also tested formulas (f) and (g) above for the functions $a(n)=\lambda(n)$, $a(n)=\sigma_0(n)$, $a(n)=\Lambda(n)$, and $a(n)=\frac{\Lambda(n)}{\log n}$ associated with the Dirichlet series for $\frac{\zeta(2s)}{\zeta(s)}$, $\zeta(s)^2$, $\frac{\zeta'(s)}{\zeta(s)}$, and $\log\zeta(s)$, and to my surprise both formulas seem to evaluate correctly for these cases as well.
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Assuming the function $G_a(s)$ defined in formula (j) below converges, it can perhaps be used to derive formulas for functions like $F_a(s)=\zeta(s)$ and $F_a(s)=\frac{\zeta'(x)}{\zeta(s)}$ associated with $a(n)=1$ and $a(n)=\Lambda(n)$ from relationships like those illustrated in formulas (k) and (l) below.
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$$G_a(s)=s\int\limits_1^\infty\tilde{f}(x)\,x^{-s-1}\,dx=\underset{\substack{K,f\to\infty \\ K\gg f\,x}}{\text{lim}}\left(-4 f \sum\limits_{k=1}^K \frac{(-1)^k}{(2 k+1)}\frac{s}{(s-2 k-1)}\sum\limits_{j=1}^k \frac{(-1)^j\ (2 \pi f)^{2(k-j)}\ F_a(2 j)}{(2 k-2 j+1)!}\right)\tag{j}$$
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$$\zeta(s)=\frac{s}{s-1}+s\int_1^\infty (\lfloor x\rfloor-x)\ x^{-s-1}\ dx\,,\quad\Re(s)>0\tag{k}$$
$$\frac{\zeta'(s)}{\zeta(s)}=\frac{s}{1-s}-s\int_1^\infty (\psi(x)-x)\ x^{-s-1}\ dx\,,\quad\Re(s)>\frac{1}{2}\quad\text{(assuming RH)}\tag{l}$$
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The following figures illustrate formula (f) for $\tilde{f}_a(x)$ above for a variety of arithmetic functions in an attempt to lend some credence to the validity of formulas (f) and (g) above. In each figure below, the reference function is illustrated in blue and formula (f) for $\tilde{f}_a(x)$ is illustrated in orange, green, and red corresponding to the evaluation frequencies $f=1$, $f=2$, and $f=3$ respectively. The red discrete evaluation points at integer values of $x$ in the figures below represent the evaluation of $f_{a_o}(x)=\frac{1}{2}(f_a(x-\epsilon)+f_a(x-\epsilon))$. The evaluation limit $K=100$ was used in all figures below.
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All figures below share a few common characteristics. First, note that in all cases the evaluation at $f=1$ (shown in orange) seems to converge to $f_{a_o}(x)$ at integer values of $x$. Second, note that in all cases increasing the evaluation frequency decreases the range of convergence with respect to $x$, but improves the fidelity of the approximation to $f_{a_o}(x)$ before the evaluation starts to diverge. The evaluation of each frequency was terminated slightly after it started to diverge in the figures below to prevent it from corrupting the rest of the plot range.
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[![Illustration of Floor(x)][3]][3]
**Figure (1)**: Illustration of $f_a(x)=\lfloor x\rfloor$ where $F_a(s)=\zeta(s)$
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[![Illustration of fa(x) associated with eta(s)][4]][4]
**Figure (2)**: Illustration of $f_a(x)=\sum\limits_{n\le x}(-1)^{n-1}$ where $F_a(s)=\eta(s)$
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[![Illustration of theta(x-1)][5]][5]
**Figure (3)**: Illustration of $f_a(x)=\theta(x-1)$ where $F_a(s)=1$
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[![Illustration of M(x)][6]][6]
**Figure (4)**: Illustration of $f_a(x)=M(x)$ ([Mertens function][7]) where $F_a(s)=\frac{1}{\zeta(s)}$
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[![Illustration of D(x)][8]][8]
**Figure (5)**: Illustration of $f_a(x)=D(x)$ ([divisor summatory function][9]) where $F_a(s)=\zeta(s)^2$
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[![Illustration of L(x)][12]][12]
**Figure (6)**: Illustration of $f_a(x)=L(x)$ ([summatory Liouville function][11]) where $F_a(s)=\frac{\zeta(2s)}{\zeta(s)}$
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[![Illustration of psi(x)][10]][10]
**Figure (7)**: Illustration of $f_a(x)=\psi(x)$ ([second Chebyshev function][13]) where $F_a(s)=\frac{-\zeta'(s)}{\zeta(s)}$
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[![Illustration of Pi(x)][14]][14]
**Figure (8)**: Illustration of $f_a(x)=\Pi(x)$ ([Riemann prime counting function][15]) where $F_a(s)=\log\zeta(s)$
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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.
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$\quad a(n)=1 \text{ where } F_a(s)=\zeta(s)$:
$$\tilde{f}_a'(x)=\sum\limits_n\delta(x-n)=\underset{f\to\infty}{\text{lim}}\left(-\frac{\sin (2 f \pi x)}{\pi x}+\sum\limits_{n=1}^f (\cos(2 n \pi x)+\cos(2 (n-1) \pi x))\right)\tag{m}$$
$$\tilde{f}_a(x)=\sum\limits_n\theta(x-n)=\underset{f\to\infty}{\text{lim}}\left(-\frac{\text{Si}(2 f \pi x)}{\pi}+\sum\limits_{n=1}^f \left(\frac{\sin(2 n \pi x)}{2 n \pi}+x\ \text{sinc}(2 (n-1) \pi x)\right)\right)\tag{n}$$
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$\quad a(n)=(-1)^{n-1} \text{ where } F_a(s)=\eta(s)$:
$$\tilde{f}_a'(x)=\sum\limits_n (-1)^{n-1}\delta(x-n)=\underset{f\to\infty}{\text{lim}}\left(\frac{\sin(2 f \pi x)}{\pi x}-2 \sum\limits_{n=1}^f \cos((2 n-1) \pi x)\right)\tag{o}$$
$$\tilde{f}_a(x)=\sum\limits_n (-1)^{n-1}\theta(x-n)=\underset{f\to\infty}{\text{lim}}\left(\frac{\text{Si}(2 f \pi x)}{\pi }-\frac{2}{\pi}\sum\limits _{n=1}^f \frac{\sin ((2 n-1) \pi x)}{2 n-1}\right)\tag{p}$$
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$\quad a(n)=\delta_{n-1} \text{ where } F_a(s)=1$:
$$\tilde{f}_a'(x)=\delta(x-1)=\underset{f\to\infty}{\text{lim}}\left(\frac{\sin(2 f \pi (x+1))}{\pi (x+1)}+\frac{\sin(2 f \pi (x-1))}{\pi (x-1)}\right)\tag{q}$$
$$\tilde{f}_a(x)=\theta(x-1)=\underset{f\to\infty}{\text{lim}}\left(\frac{\text{Si}(2 f \pi (x+1))+\text{Si}(2 f \pi (x-1))}{\pi }\right)\tag{r}$$
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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 $\Re(s)>-1$.
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$$\zeta(s)=\underset{f\to\infty}{\text{lim}}\left((2 \pi )^{s-1} \sin \left(\frac{\pi s}{2}\right)\ \Gamma (1-s) \left(-\frac{2 f^s}{s}+\sum _{n=1}^f \left(n^{s-1}+(n-1)^{s-1}\right)\right)\right),\ 1<\Re(s)<2\tag{s}$$
$$\eta(s)=\underset{f\to\infty}{\text{lim}}\left(2 \pi^{s-1} \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s) \left(\frac{2^{s-1} f^s}{s}-\sum\limits_{n=1}^f (2 n-1)^{s-1}\right)\right),\ \Re(s)<2\tag{t}$$
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Formulas (n), (p), (r), and (t) above are illustrated in this [answer][16] I recently posted to my own question on MathOverflow.
[1]: https://i.sstatic.net/NIhrQ.jpg
[2]: https://i.sstatic.net/WFLiZ.jpg
[3]: https://i.sstatic.net/yilEU.jpg
[4]: https://i.sstatic.net/CQ2wA.jpg
[5]: https://i.sstatic.net/3yDa8.jpg
[6]: https://i.sstatic.net/l36U1.jpg
[7]: https://en.wikipedia.org/wiki/Mertens_function
[8]: https://i.sstatic.net/iGtXd.jpg
[9]: https://en.wikipedia.org/wiki/Divisor_summatory_function
[10]: https://i.sstatic.net/P7J9i.jpg
[11]: https://en.wikipedia.org/wiki/Liouville_function
[12]: https://i.sstatic.net/N8Vye.jpg
[13]: https://en.wikipedia.org/wiki/Chebyshev_function
[14]: https://i.sstatic.net/ll0r0.jpg
[15]: https://mathworld.wolfram.com/RiemannPrimeCountingFunction.html
[16]: https://mathoverflow.net/q/395266