This question is related to the three functions defined in (1) to (3) below where $\coth(z)$ gives the hyperbolic cotangent of $z$.
(1) $\quad M(x)=\sum\limits_{n=1}^x\mu(n)\quad\text{(Mertens function)}$
(2) $\quad\frac{1}{e}=\frac{1}{2}\sum\limits_{n=1}^N\mu(n)\,\coth\left(\frac{n}{2}\right)\,,\quad M(N)=0\land N\to\infty$
(3) $\quad f(s)=\frac{1}{2}\sum\limits_{n=1}^N\mu(n)\,\coth\left(\frac{n}{2}\right)\,n^{-s}\,,\quad N\to\infty$
The following three figures illustrate the error in formula (2) for $\frac{1}{e}$ as a function of the upper evaluation limit $N$. The red discrete portions of the figures illustrate the error in formula (2) when $M(N)=0$. Formula (2) converges fairly rapidly (when $M(N)=0$), e.g. the error in formula (2) evaluated at $N=39$ is approximately $-2.343\times 10^{-18}$.
Figure (1): Error in Formula (2) for $\frac{1}{e}$ for $1\le N\le 100$
Figure (2): Error in Formula (2) for $\frac{1}{e}$ for $1\le N\le 1000$
Figure (3): Error in Formula (2) for $\frac{1}{e}$ for $1\le N\le 10000$
The following figure illustrates a discrete plot of $\coth\left(\frac{n}{2}\right)$ (related to formulas (2) and (3) above) which converges fairly rapidly to $1$ as $n\to\infty$ which explains why the evaluation of formula (2) for $\frac{1}{e}$ changes very little between successive values of $N$ for which $M(N)=0$ as $N\to\infty$.
Figure (4): $\coth\left(\frac{n}{2}\right)$
The following three figures illustrate the absolute value, real, and imaginary parts of formula (3) for $f(s)$ evaluated along the critical line ($s=1/2+i\,t$) using an upper evaluation limit of $N=10,000$. The red discrete portions of the three figures below illustrate the evaluation of formula (3) at the first $10$ non-trivial zeta zeros in the upper-half plane.
Figure (5): Evaluation of $\left|f\left(\frac{1}{2}+i\,t\right)\right|$
Figure (6): Evaluation of $\Re\left(f\left(\frac{1}{2}+i\,t\right)\right)$
Figure (7): Evaluation of $\Im\left(f\left(\frac{1}{2}+i\,t\right)\right)$
Question (1): What is the Dirichlet transform of $a(n)=\mu(n)\,\coth\left(\frac{n}{2}\right)$? In particular, can $f(s)$ defined in formula (3) above be expressed in closed-form in terms of the Riemann zeta function $\zeta(s)$?