I've been exploring recovering Riemann's prime-power counting function $J[y]$ and the second Chebyshev function $\psi(y)$ from functions of $\zeta[s]$ via relationships such as the following.
$$J(y)=\frac{1}{2\pi i}\int_{a-\infty\ i}^{a+\infty\ i}\log\zeta[s]\ y^s\frac{ds}{s}$$
$$\psi(y)=\frac{1}{2\pi i}\int_{a-\infty\ i}^{a+\infty\ i}(-\frac{\zeta'[s]}{\zeta[s]})\ y^s\frac{ds}{s}$$
I've noticed these relationships seem to evaluate to twice the step-size of the associated prime counting function, and I'm trying to understand if there's an error in these relationships or an error in my evaluation method.
I'm primarily interested in investigating these relationships in the context of using Fourier series representations of $J'[x]$ and $\psi'[x]$ to recover $\log\zeta[s]$ and $-\frac{\zeta'[s]}{\zeta[s]}$ respectively (see Illustrations of Fourier Series for Prime Counting Functions), but below I illustrate the problem I'm encountering in a simpler context.
For example, the second relationship above can be evaluated using term-wise integration as follows. This approach is based on section 3.2 of Edward's book "Riemann's Zeta Function".
$$-\frac{\zeta'[s]}{\zeta[s]}=\int_{0}^{\infty}\psi'[x]\ x^{-s}dx=\sum_{n=2}^\infty \Lambda[n]\ n^{-s}$$
$$\psi(y)=\sum_{n=2}^\infty \Lambda[n]\frac{1}{2\pi i}\int_{a-\infty\ i}^{a+\infty\ i}(\frac{y}{n})^s\ \frac{ds}{s}$$
I believe the integral above evaluates as follows.
$$\int(\frac{y}{n})^s\ \frac{ds}{s}=Ei[s\log\frac{y}{n}]$$
This leads to the following formula to recover $\psi[y]$. $$\psi(y)=\frac{1}{2\pi i}\sum_{n=2}^N \Lambda[n]\ (Ei[(a+i\ M)\log\frac{y}{n}]-Ei[(a-i\ M)\log\frac{y}{n}]),\quad N,M\to\infty$$
The formula above evaluates with an offset which I'm less concerned about and which can can be taken care of by subtracting off the evaluation at y=1. But as I noted earlier, the formula above seems to evaluate to twice the step size of $\psi[y]$ which I'm more concerned about.
So my question is:
Are the relationships above correct or is there an error in my evaluation method? For example, are these relationships conditionally convergent based on assumed relationships between the relative values of the evaluation parameters N, a, and M?
Another discrepancy which I've noticed is the following integral is supposed to evaluate to zero for $0<z<1,$ and it seems to evaluate to $-1$ for $0<z<1$, which is most likely related to the evaluation problems I'm having with the relationships above.
$$\frac{1}{2\pi i}\int_{a-\infty\ i}^{a+\infty\ i} z^s\frac{ds}{s}$$