This question assumes definition (1) below and relationship (2) below. With respect to the integral in (2) below, I selected $\frac{1}{2}$ as the lower integration bound because this is the ideal location for minimizing the undesirable contribution of the step of $S(x)$ at $x=0$ while simultaneously maximizing the desirable contribution of the step of $S(x)$ at $x=1$.
(1) $\quad S(x)=x-\left(\frac{1}{2}-\frac{1}{\pi}\sum\limits_{k=1}^f\frac{\sin(2\,\pi\,k\,x)}{k}\right),\quad f\to\infty$
(2) $\quad\zeta(s)=s\int\limits_{1/2}^\infty S(x)\,x^{-s-1}\,dx$
I originally illustrated a couple of formulas for $\zeta(s)$ based on definition (1) and relationship (2) above in my earlier question Are these formulas for the Riemann zeta function $\zeta(s)$ globally convergent? which involved the hypergeometric $_1F_2$ function.
The question here is about formula (3) below which was also derived from definition (1) and relationship (2) above but is also based on this answer to my follow-on question What is $s\int_1^\infty\sin(2\,\pi\,n\,x)\,x^{-s-1}\,dx$?
(3) $\quad\zeta(s)=\underset{f\to\infty}{\text{lim}}\quad 2^{\,s-1}\left(\frac{s}{s-1}-1+\sum\limits_{n=1}^f\left(E_s(i n \pi)+E_s(-i n \pi)\right)\right)$
Formula (3) above for $\zeta(s)$ is illustrated following the questions below.
Question (1): Is formula (3) for $\zeta(s)$ above globally convergent as $f\to\infty$?
Question (2): If so, does global convergence of formula (3) for $\zeta(s)$ have any implications with respect to the Riemann Hypothesis?
Question (3): If not, what is the convergence range of this formula?
Formula (5) below defines another globally convergent formula for $\zeta(s)$ based on relationship (4) below and this second answer to my question What is $s\int_1^\infty\sin(2\,\pi\,n\,x)\,x^{-s-1}\,dx$?.
(4) $\quad\zeta(s)=s\int\limits_1^\infty S(x)\,x^{-s-1}\,dx$
(5) $\quad\zeta(s)=\underset{K\to\infty}{\text{lim}}\left(\frac{s}{s-1}-\frac{1}{2}+\sum\limits_{n=1}^K\left((2 \pi i n)^{s-1} \Gamma (1-s,2 \pi i n)+(-2 \pi i n)^{s-1} \Gamma (1-s,-2 \pi i n)\right)\right)\\$ $\qquad\qquad\quad=\underset{K\to\infty}{\text{lim}}\left(\frac{s}{s-1}-\frac{1}{2}+\sum_\limits{n=1}^K\left(E_s(2 \pi i n)+E_s(-2 \pi i n)\right)\right)$
Note formula (3) for $\zeta(s)$ above was derived from the relationship $\zeta(s)=s\int\limits_{1/2}^\infty S(x)\,x^{-s-1}\,dx$ and formula (5) for $\zeta(s)$ above was derived from the relationship $\zeta(s)=s\int\limits_1^\infty S(x)\,x^{-s-1}\,dx$.
Question (4): Can a globally convergent formula for $\zeta(s)$ be derived from the more general integral $\zeta(s)=s\int\limits_a^\infty S(x)\,x^{-s-1}\,dx$ for any $0<a\le 1$?
The following figure illustrate formula (3) for $\zeta(s)$ in orange where formula (3) is evaluated with the upper limit $f=20$. The underlying blue reference function is $\zeta(s)$.
Figure (1): Illustration of formula (3) for $\zeta(s)$ evaluated at $f=20$
The following four figures illustrate the absolute value, real part, imaginary part, and argument of formula (3) for $\zeta(s)$ evaluated along the critical line $s=\frac{1}{2}+i\,t$ in orange where formula (3) is evaluated with the upper limit $f=20$. The underlying blue reference function is $\zeta(\frac{1}{2}+i\,t)$. The red discrete portion of the plot illustrates the evaluation of formula (3) at the first $10$ non-trivial zeta-zeros in the upper half-plane.
Figure (2): Illustration of formula (3) for $\left|\zeta\left(\frac{1}{2}+i\,t\right)\right|$ evaluated at $f=20$
Figure (3): Illustration of formula (3) for $\Re\left(\zeta\left(\frac{1}{2}+i\,t\right)\right)$ evaluated at $f=20$
Figure (4): Illustration of formula (3) for $\Im\left(\zeta\left(\frac{1}{2}+i\,t\right)\right)$ evaluated at $f=20$
Figure (5): Illustration of formula (3) for $\text{Arg}\left(\zeta\left(\frac{1}{2}+i\,t\right)\right)$ evaluated at $f=20$