When I was combining the identities from this article from Wikipedia for the Mertens function, I've asked my an open question, if you can solve it from a standard viewpoint it is appreciated, and should be a nice exercise to learn basics about these functions and methods in analytic number theory.
Question. Are there some substancials differences between the second Chebyshev function $\psi(x)=\sum_{n\leq x}\Lambda(n)$ (where thus $\Lambda(n)$ is the von Mangoldt function) and this different function $$\phi(x):=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\eta'(s)\frac{x^s}{s\zeta(s)}ds,\tag{1}$$ for $c>1$ where we are denoting with $\eta(s)$ the alternating zeta function (see in this Wikipedia)?
Note the similarity of my definition because I wrote the factor $\eta'(s)$ instead of the derivative of the Riemann's Zeta function. With the words substancials differences I am asking if you can tell us how trackle and write (if it is feasible) the corresponding definition of $\phi(x)$ as a step function. That is, can you provide me a discussion about similarities and differences of the new and old step functions? I say the more important facts, see also below if you want to expand the answer. Thanks in advance.
$$\eta(s) = (1-2^{1-s})\zeta(s), \qquad \eta'(s) = (1-2^{1-s}) \zeta'(s)+ \ln 2 \ 2^{1-s} \zeta(s)$$ For $ Re(s) > 1$ : $$\frac{\eta'(s)}{\zeta(s)} = s\int_1^\infty (\psi(x)-2\psi(x/2)+ 2\ln 2 \ 1_{x > 2} )x^{-s-1}dx$$ By inverse Mellin/Laplace/Fourier transform, for $c > 1$ and $x > 0$ : $$\psi(x)-2\psi(x/2)+ 2\ln 2 \ 1_{x > 2}=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\eta'(s)}{\zeta(s)}\frac{x^s}{s}ds$$ $\scriptstyle\text{as usual with Fourier transforms, at the discontinuities } (x \in \mathbb{N}) \text{ we define } \psi(x) \text{ to be the mean value of its left and right limits}$