This question is related to the prime-counting functions defined in formulas (1) to (4) below
$$A(x)=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\log(n)^2}\tag{1}$$
$$\Pi(x)=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\log(n)}\tag{2}$$
$$\psi(x)=\sum\limits_{n=1}^x\Lambda(n)\tag{3}$$
$$B(x)=\sum\limits_{n=1}^x\Lambda(n)\log(n)\tag{4}$$
where $\Pi(x)$ is Riemann's prime-power counting function and $\psi(x)$ is the second Chebyshev function.
The prime-counting functions defined in (1) to (4) above are related via their first-order derivatives as follows.
$$\Pi'(x)=\log(x)\,A'(x)\tag{5}$$
$$\psi'(x)=\log(x)\,\Pi'(x)\tag{6}$$
$$B'(x)=\log(x)\,\psi'(x)\tag{7}$$
The asymptotics for the prime counting functions defined in (1) to (4) above are as follows.
$$A(x)\approx li(x)-\frac{x}{\log(x)}\tag{8}$$
$$\Pi(x)\approx li(x)\tag{9}$$
$$\psi(x)\approx x\tag{10}$$
$$B(x)\approx x\,(\log(x)-1)\tag{11}$$
I know the error bounds on $|\Pi(x)-li(x)|$ and $|\psi(x)-x|$ predicted by the Prime Number Theorem (PNT) and Riemann Hypothesis (RH) have been thoroughly researched and are well documented, but I'm wondering about $A(x)$ and $B(x)$.
Question (1): What are the error bounds for $|A(x)-(li(x)-\frac{x}{\log(x)})|$ predicted by the PNT and RH?
Question (2): What are the error bounds for $|B(x)-(x\,(\log(x)-1))|$ predicted by the PNT and RH?
Note the asymptotic $li(x)-\frac{x}{\log(x)}$ for the prime counting function $A(x)$ is the difference between the two most famous estimates for the asymptotic for the fundamental prime counting function $\pi(x)$.
The prime counting functions defined in (1) to (4) above are related the Riemann zeta function $\zeta(s)$ as follows.
$$\int\log\zeta(s)\,ds=-s\int\limits_0^\infty A(x)\,x^{-s-1}\,ds=-\sum\limits_{n=2}^\infty\frac{\Lambda(n)}{\log(n)^2}\,n^{-s}\,,\quad\Re(s)>1\tag{12}$$
$$\log\zeta(s)=s\int\limits_0^\infty\Pi(x)\,x^{-s-1}\,ds=\sum\limits_{n=2}^\infty\frac{\Lambda(n)}{\log(n)}\,n^{-s}\,,\quad\Re(s)>1\tag{13}$$
$$\frac{\partial\,\log\zeta(s)}{\partial s}=-s\int\limits_0^\infty\psi(x)\,x^{-s-1}\,ds=-\sum\limits_{n=1}^\infty\Lambda(n)\,n^{-s}\,,\quad\Re(s)>1\tag{14}$$
$$\frac{\partial^2\,\log\zeta(s)}{\partial s^2}=s\int\limits_0^\infty B(x)\,x^{-s-1}\,ds=\sum\limits_{n=1}^\infty\Lambda(n)\log(n)\,n^{-s}\,,\quad\Re(s)>1\tag{15}$$
Question (3): Is there a formula for $\int\log\zeta(s)\,ds$ that converges for $\Re(s)\le 1$ and if so, what is its definition?
Riemann's explicit formulas for the prime-power counting function $\Pi(x)$ and von Mangloldt's explicit formula for the second Chebyshev function $\psi(x)$ are as follows.
$$\Pi(x)=li(x)-\sum\limits_\rho Ei(\log\,(x)\,\rho)-\log(2)-\sum\limits_{n=1}^\infty Ei(-2\,n\,\log(x))\,,\quad x>1\tag{16a}$$ $$\Pi(x)=li(x)-\sum\limits_\rho Ei(\log(x)\,\rho)-\log (2)+\int_x^\infty\frac{1}{t\,\left(t^2-1\right)\log(t)}\,dt\,,\quad x>1\tag{16b}$$
$$\psi(x)=x-\sum\limits_\rho\frac{x^\rho}{\rho}-\log(2\,\pi)+\sum\limits_{n=1}^\infty\frac{x^{-2\,n}}{2\,n}\,,\quad x>1\tag{17a}$$ $$\psi(x)=x-\sum\limits_\rho\frac{x^\rho}{\rho}-\log(2\,\pi)-\frac{1}{2}\,\log\left(1-\frac{1}{x^2}\right)\,,\quad x>1\tag{17b}$$
I derived the following explicit formula for the prime counting function $B(x)$ where $Li_2$ is the polylogarithm function and $b\approx 0.63$.
$$B(x)=x\,(\log(x)-1)-\sum\limits_\rho\frac{x^\rho\,(\rho\,\log(x)-1)}{\rho^2}+b+$$ $$\sum\limits_{n=1}^\infty\frac{(2\,n\log(x)+1)\,x^{-2\,n}}{4\,n^2}\,,\quad x>1\tag{18a}$$ $$B(x)=x\,(\log(x)-1)-\sum\limits_\rho\frac{x^\rho\,(\rho\,\log(x)-1)}{\rho^2}+b+$$ $$\frac{1}{4}\left(Li_2\left(\frac{1}{x^2}\right)-2\,\log(x)\log\left(1-\frac{1}{x^2}\right)\right),\quad x>1\tag{18b}$$
The following plot illustrates the explicit formula for $B(x)$ defined in (18) above in orange and the reference function $B(x)$ defined in (4) above in blue. The explicit formula is evaluated over the first 100 pairs of zeta zeros with $b=0.63$.
$B(x)$" />I originally derived the explicit formula for $B(x)$ defined in (18) above via two different approaches which are outlined in (19) and (20) below. Both approaches start with the first explicit formula for $\psi(x)$ defined in (17) above which uses the series representation of the final error term. Both approaches provide exactly the same result, but I'm not sure how to handle the derivation of the constant offset term $b$ for either approach.
$B(x)=\int\log(x)\frac{\partial\,\psi(x)}{\partial\,x}\,dx\tag{19}$
$$B(x)=\frac{1}{2\,\pi\,\,i}\int\limits_{c-i\,\infty}^{c+i\,\infty}\frac{\partial}{\partial\,s}\left(-s\int\limits_0^\infty\psi(x)\,x^{-s-1}\,ds\right)\frac{x^s}{s}\,ds\tag{20}$$
The first approach outlined in (19) above leverages the relationship $B'(x)=\log(x)\,\psi'(x)$ from (7) above.
The second approach outlined in (20) above takes advantage of the fact that all terms in the first formula for $\psi(x)$ illustrated in (17) above are of the form $a\, x^z$ and the Mellin transform of $x^z$ is $2\,\pi\,\delta(i\,(s+z))$. I illustrated $\mathcal{M}_s^{-1}[2\,\pi\,\delta(i\,(s+z))](x)=x^z$ in the following answer I posted to a question on evaluation of $\delta(s)$ for $s\in \mathbb{C}$.
Answer to Question on Evaluation of $\delta(s)$ for $s\in \mathbb{C}$
I illustrated a single outer integral in approach 2 outlined in (20) above, but this is a bit of an oversimplification as each individual term requires a different value of $c$ for evaluation of its associated integral.
Question (4): What is the derivation and exact value for the constant offset term $b$ associated with the derived explicit formula for $B(x)$ defined in (18) above?
I derived the following explicit formula for the prime counting function $A(x)$.
$$A(x)=li(x)-\frac{x}{\log(x)}+\sum\limits_\rho\left(\rho\, E_1\left(-\rho\, \log(x)\right)+\frac{x^{\rho}}{\log(x)}\right)+a+$$ $$\sum\limits_{n=1}^\infty\left(2\,n\, Ei(-2\,n\,\log(x))+\frac{x^{-2\,n}}{\log(x)}\right),\quad x>1\tag{21a}$$ $$A(x)=li(x)-\frac{x}{\log(x)}+\sum\limits_\rho\left(\rho\, E_1\left(-\rho\, \log(x)\right)+\frac{x^{\rho}}{\log(x)}\right)+a+$$ $$\sum\limits_{n=1}^\infty (2\,n\,Ei(-2\,n\,\log(x)))+\frac{1}{\left(x^2-1\right)\,\log(x)}\,,\quad x>1\tag{21b}$$
The following plot illustrates the explicit formula for $A(x)$ defined in (21) above in orange and the reference function $A(x)$ defined in (1) above in blue. The explicit formula is evaluated over the first 100 pairs of zeta zeros with $a=2.5$.
$A(x)$" />I derived the explicit formula for $A(x)$ defined in (21) above via the approach which is outlined in (22) below. This approach starts with the first explicit formula for $\Pi(x)$ defined in (16) above which uses the series representation of the final error term. The approach outlined in (22) below leverages the relationship $\Pi'(x)=\log(x)\,A'(x)$ from (5) above. I'm not sure how to handle the derivation of the constant offset term $a$ for this approach.
$A(x)=\int\frac{1}{log(x)}\frac{\partial\,\Pi(x)}{\partial\,x}\,dx\tag{22}$
Question (5): What is the derivation and exact value for the constant offset term $a$ associated with the derived explicit formula for $A(x)$ defined in (21) above?
March 17, 2018 Update:
I believe I've determined the answer to question (4) above. I believe the constant offset term $b$ associated with the derived explicit formula for $B(x)$ defined in (18) above is
$$b=\frac{\zeta''(0)}{\zeta(0)}-\frac{\zeta'(0)^2}{\zeta(0)^2}=\frac{\pi ^2}{12}-\gamma^2-2 \gamma_1=0.634921\tag{23}$$
which corresponds to the residue of $\frac{\partial^2\,\log\zeta(s)}{\partial s^2} \frac{x^s}{s}$ at $s=0$.
I more recently verified the remainder of the terms in formula (18) for $B(x)$ above are consistent with the residues of $\frac{\partial^2\,\log\zeta(s)}{\partial s^2} \frac{x^s}{s}$ at $s=1$ and the zeros of $\zeta(s)$.
In the explicit formulas defined above, the sums $\sum\limits_{\rho}\ (...)$ over the non-trivial zeta zeros are conditionally convergent and should be evaluated as $\underset{T\to\infty}{\text{lim}}\left(\sum\limits_{|\Im(\rho)|<T}\ (...)\right)$.