Calculate a good approximation of $\lim_{x\to\infty}\int_2^x e^{-\pi(t)}dt$, where $\pi(x)$ is the prime-counting function

Question

Inspired in the definition of the Golomb–Dickman constant, see for example in page 529 of [1] (free access from the web of the AMS), I consider $$\int_2^x e^{-\pi(t)}dt,$$ where $\pi(x)$ is the prime-counting function, see its definition for example from this MathWorld.

Question. How can we calculate a good approximation of $$\int_2^\infty e^{-\pi(x)}dx?$$ Many thanks.

References:

[1] Lagarias, Euler's constant: Euler's work and modern developments, Bulletin of the American Mathematical Society, 50 (4), (2013).

Answer 1

Numerically, this does not seem to make any problem if we consider $$f_n=\int_{p_n}^{p_{n+1}} e^{-\pi (x)} \, dx\qquad \text{and}\qquad g_m=\sum_{n=1}^m f_n=\sum_{n=1}^{m} (p_{n+1}-p_n)\, e^{-n}$$ (see Antonio Vargas comment).

Below is a table of generated values $$\left( \begin{array}{cc} m & g_m \\ 5 & 0.82486259393350324671035177334581504486982084749638 \\ 10 & 0.83877447576693280717388127491652502994294656859973 \\ 15 & 0.83890894500870098130370987262554128951261245468174 \\ 20 & 0.83890982093187513586368658110075745562346018599660 \\ 25 & 0.83890982757047394666269786169926490186502317042721 \\ 30 & 0.83890982759192065348095727852873887161463474591048 \\ 35 & 0.83890982759216205543854829896368181611163602891794 \\ 40 & 0.83890982759216417947988624151469749806143841461256 \\ 45 & 0.83890982759216418914669475325279067175180058855706 \\ 50 & 0.83890982759216418932707086649050466597792079421674 \\ 55 & 0.83890982759216418932767382058216343894327616862031 \\ 60 & 0.83890982759216418932767754181410766698373920809439 \\ 65 & 0.83890982759216418932767759291902842598347660179781 \\ 70 & 0.83890982759216418932767759330393721260699403750457 \\ 75 & 0.83890982759216418932767759330541827711774044359378 \\ 80 & 0.83890982759216418932767759330542819614193086258447 \\ 85 & 0.83890982759216418932767759330542823811431792371867 \\ 90 & 0.83890982759216418932767759330542823854636033633091 \\ 95 & 0.83890982759216418932767759330542823855117010762525 \\ 100 & 0.83890982759216418932767759330542823855119385370828 \\ 105 & 0.83890982759216418932767759330542823855119403474903 \\ 110 & 0.83890982759216418932767759330542823855119403596887 \\ 115 & 0.83890982759216418932767759330542823855119403597417 \\ 120 & 0.83890982759216418932767759330542823855119403597418 \end{array} \right)$$

Inverse symbolic calculators do not find anything looking like this number.

Edit

What looks to be interesting is that, computed up to $m=5000$, a simple linear regression leads to $$\log_{10}(g_{m+1}-g_m)=-0.434167\, m$$ showing that the $50$ significant figures are obtained for $m=116$ (as shown by the table) and that $100$ significant figures are obtained for $m=231$ (which has been checked).