The impact of $\zeta(s)$ complex zeros on $\pi(x)$ for $x<10^{10^{13}}$

An analysis of the local variations of the prime counting function $\pi(x)$ due to the impact of the non-trivial, complex zeros $\varrho_k$ of $\zeta(s)$ is provided for $x<10^{10^{13}}$ using up to 200 billion $\zeta(s)$ complex zeros. A new bound for $\vert\mathrm{li}(x)-\pi(x)\vert<x^{1/2}(\mathrm{log} \mathrm{log} \mathrm{log} x+e+1)/e \mathrm{log} x$ is proposed consistent with the error growth rate in Littlewood's proof that $\mathrm{li}(x)-\pi(x)$ changes sign infinitely often. This bound is also consistent with all presently known cases where $\pi(x)>\mathrm{li}(x)$ including many new examples listed. This implies that Littlewood's constant $\mathrm{K}=1/e$, the lower bound for Skewes' number is $3.17\times 10^{114}$ and the positive constant $c$ in the Riemann Hypothesis equivalent $\vert\mathrm{li}(x)-\pi(x)\vert<c \mathrm{log}(x)x^{1/2}$ is less than $3\times 10^{-27}$.