Relationships (1) and (2) below are valid for $\Re(s)>1$.
(1) $\quad\int_1^\infty\psi(x)\ x^{-s-1}\ dx=-\frac{\zeta'(s)}{s\ \zeta(s)}\,,\quad \Re(s)>1$
(2) $\quad\int_1^\infty\psi'(x)\ x^{-s}\ dx=-\frac{\zeta'(s)}{\ \zeta(s)}\,,\quad \Re(s)>1$
For $\Re(s)>1$, relationship (3) below follows from relationship (1) above and relationship (4) below. I've been told the convergence of (3) for $Re(s)=1$ is equivalent to the Prime Number Theorem, and convergence of (3) for $Re(s)\in\left(\frac{1}{2},1\right)$ is consistent with the Riemann Hypothesis.
(3) $\quad\int_1^\infty(\psi(x)-x)\ x^{-s-1}\ dx=-\frac{\zeta'(s)}{s\ \zeta(s)}+\frac{1}{1-s}\,,\quad \Re(s)>\frac{1}{2}\ (assuming\ RH)$
(4) $\quad\int_1^\infty (-x)\ x^{-s-1}\ dx=\frac{1}{1-s}\,,\quad \Re(s)>1$
For $\Re(s)>1$, relationship (5) below follows from relationship (2) above and relationship (6) below. I believe the convergence of (5) for $Re(s)=1$ is equivalent to the Prime Number Theorem, and convergence of (5) for $Re(s)\in\left(\frac{1}{2},1\right)$ is consistent with the Riemann Hypothesis.
(5) $\quad\int_1^\infty(\psi'(x)-1)\ x^{-s}\ dx=-\frac{\zeta'(s)}{\ \zeta(s)}+\frac{1}{1-s}\,,\quad \Re(s)>\frac{1}{2}\ (assuming\ RH)$
(6) $\quad\int_1^\infty (-1)\ x^{-s}\ dx=\frac{1}{1-s}\,,\quad \Re(s)>1$
I believe that for $Re(s)\in\left(\frac{1}{2},1\right)$, relationship (3) above follows from relationships (7) and (8) below by splitting the integral in (7) at $x=1$ and moving the lower portion of the integral to the right side of (7).
(7) $\quad\int_0^\infty(\psi(x)-x)\ x^{-s-1}\ dx=-\frac{\zeta'(s)}{s\ \zeta(s)}\,,\quad \Re(s)\in\left(\frac{1}{2},1\right)\ (assuming\ RH)$
(8) $\quad\int_0^1 (-x)\,x^{-s-1}\,dx=-\frac{1}{1-s}\,,\quad \Re(s)<1$
I believe that for $Re(s)\in\left(\frac{1}{2},1\right)$, relationship (5) above follows from relationships (9) and (10) below by splitting the integral in (9) at $x=1$ and moving the lower portion of the integral to the right side of (9).
(9) $\quad\int_0^\infty(\psi'(x)-1)\ x^{-s}\ dx=-\frac{\zeta'(s)}{\ \zeta(s)}\,,\quad \Re(s)\in\left(\frac{1}{2},1\right)\ (assuming\ RH)$
(10) $\quad\int_0^1 (-1)\,x^{-s}\ dx=-\frac{1}{1-s}\,,\quad \Re(s)<1$
I'm been attempting to validate (3) and (5) above using (11) and (12) below and this seems to work well for $Re(s)\ge 1$, but not so well for $Re(s)\in\left(\frac{1}{2},1\right)$. I've noticed (11) and (12) seem to converge for $Re(s)\ge1$, but I don't believe either (11) or (12) converge for $Re(s)\in\left(\frac{1}{2},1\right)$.
(11) $\quad\int_1^\infty(\psi(x)-x)\ x^{-s-1}\ dx=\sum _{n=1}^N \frac{\Lambda (n) \left(n^{-s}-N^{-s}\right)}{s}+\frac{1-N^{1-s}}{1-s}\,,\quad\Re(s)\ge 1\ \&\ N\to\infty$
(12) $\quad\int_1^\infty(\psi'(x)-1)\ x^{-s}\ dx=\sum_{n=1}^N\Lambda (n)\,n^{-s}+\frac{1-N^{1-s}}{1-s}\,,\quad \Re(s)\ge 1\ \&\ N\to\infty$
The following three plots illustrate the error between the right side of (11) and the right side of (3) for $\Re(s)=2$, $\Re(s)=1$, and $\Re(s)=\frac{1}{2}+0.01$ using evaluation limits of $N=100$ (blue) and $N=1000$ (orange). Note as $N$ increases from $100$ (blue) to $1000$ (orange), the right side of (11) seems to be converging to the right side of (3) in the first plot for $\Re(s)=2$ and in the second plot for $\Re(s)=1$, but not in the third plot for $\Re(s)=\frac{1}{2}+0.01$.
The following three plots illustrate the error between the right side of (12) and the right side of (5) for $\Re(s)=2$, $\Re(s)=1$, and $\Re(s)=\frac{1}{2}+0.01$ using evaluation limits of $N=100$ (blue) and $N=1000$ (orange). Note as $N$ increases from $100$ (blue) to $1000$ (orange), the right side of (12) seems to be converging to the right side of (5) in the first plot for $\Re(s)=2$ and in the second plot for $\Re(s)=1$, but not in the third plot for $\Re(s)=\frac{1}{2}+0.01$.
Question 1: Assuming the Riemann Hypothesis, is the right side of (11) supposed to converge to the right side of (3) as $N\to\infty$ for $Re(s)\in\left(\frac{1}{2},1\right)$?
Question 2: If the answer to question 1 is no, what formula can be used to evaluate the integral on the left side of (11) for $Re(s)\in\left(\frac{1}{2},1\right)$?
Question 3: Assuming the Riemann Hypothesis, is the right side of (12) supposed to converge to the right side of (5) as $N\to\infty$ for $Re(s)\in\left(\frac{1}{2},1\right)$?
Question 4: If the answer to question 3 is no, what formula can be used to evaluate the integral on the left side of (12) for $Re(s)\in\left(\frac{1}{2},1\right)$?
With respect to questions 2 and 4 above, I'm interested in formulas derived from the $\Lambda(n)$, $UnitStep(x-n)$, and $\delta(x-n)$ functions versus formulas derived from von Mangoldt's explicit formula involving a sum over the zeta zeros.
