This question is related to the prime-counting function $\Pi(x)$ and staircase function $Q(x)$ defined in (1) and (2) below respectively.
(1) $\quad\Pi(x)=\sum\limits_{n=2}^\infty\frac{\Lambda(n)}{\log(n)}\,\theta(x-n)\qquad\qquad\qquad\text{(Riemann's Prime-Power Counting Function)}$
(2) $\quad Q(x)=\sum\limits_{n=2}^\infty\frac{1}{\log(n)}\,\theta(x-n)-\delta (x-1)$
The asymptotics for $\Pi(x)$ and $Q(x)$ are as follows.
(3) $\quad\Pi(x)\approx\Pi_a(x)=li(x)-\log(2)$
(4) $\quad Q(x)\approx Q_a(x)=li(x)-c\,,\quad c\approx\frac{1}{4}$
The staircase function $Q(x)$ defined in (2) above has a fairly narrow and precise error bound with respect to its asymptotic $Q_a(x)$ which is as follows.
(5) $\quad|Q(x)-Q_a(x)|\le\frac{1}{2\,\log(x)}$
The following two plots illustrate $Q(x)-Q_a(x)$ in blue evaluated with $c=\frac{1}{4}$ and the error bounds $-\frac{1}{2\,\log(x)}$ and $\frac{1}{2\,\log(x)}$ in dashed-red. The second plot below illustrates $c=\frac{1}{4}$ is close but not exactly correct.
Question (1): What is the exact value of $c$ associated with the asymptotic for $Q_a(x)$ defined in (4) above consistent with the error bound for $Q(x)$ defined in (5) above?
Note the asymptotics for Riemann's prime-power counting function $\Pi(x)$ and the staircase function $Q(x)$ are both essentially $li(x)$ ignoring the minor constants $-\log(2)$ associated with $\Pi(x)$ and $a$ associated with $Q(x)$. The following plot illustrates $\Pi(x)$ (blue), $Q(x)$ (orange), and the asymptotic $li(x)$ (gray-dashed curve).
Question (2): Does the asymptotic $Q_a(x)$ and associated error bound $|Q(x)-Q_a(x)|\le\frac{1}{2\,\log(x)}$ for the staircase function $Q(x)$ provide any meaningful insight into the Riemann hypothesis?
The staircase function $Q(x)$ defined in (2) above is related the Riemann zeta function $\zeta(s)$ as follows.
(6) $\quad\int\zeta(s)\,ds=-s\int\limits_0^\infty Q(x)\,x^{-s-1}\,ds=s-\sum\limits_{n=2}^\infty\frac{1}{\log(n)}\,n^{-s}\,,\quad\Re(s)>1$
Question (3): Is there a functional equation for $\int\zeta(s)\,ds$ and if so, what is its definition? If not, is there a formula for $\int\zeta(s)\,ds$ that converges for $\Re(s)\le 1$ and if so, what is its definition?