There have been many results on the first sign change of $\pi(x)-{\mathrm{li}}(x)$: among others, Lehman, te Riele, Bays & Hudson, Demichael, Chao & Plymen, and most recently Saouter & Demichel. These provide upper bounds (as well as lower bounds on some region in which $\pi(x)>{\mathrm{li}}(x)$).

Could these methods be used to generate a lower bound? That is, a region [2, y] in which $\pi(x)<{\mathrm{li}}(x)$. (Or even a region [x, y] where y is smaller than the best known upper bound, such that, in principle at least, direct calculations could yield a lower bound.)

It seems unlikely that direct searches like [2] will ever be able to resolve the exact value of the first crossing.

[1] C. Bays and R. H. Hudson. "A new bound for the smallest x with $\pi(x)>{\mathrm{li}}(x)$" *Mathematics of Computation* **69** (2000), pp. 1285–1296.

[2] T. Kotnik. "The prime-counting function and its analytic approximations", *Advances in Computational Mathematics* **29** (2008), pp. 55-70.

[2] R. Sherman Lehman. "On the difference $\pi(x)-{\mathrm{li}}(x)$" *Acta Arithmetica* **11** (1965), pp. 397–410.

[3] H. J. J. te Riele. "On the sign of the difference $\pi(x)-{\mathrm{li}}(x)$" *Mathematics of Computation* **48** (1987), pp. 323–328.

[4] Yannick Saouter and Patrick Demichel. "A sharp region where $\pi(x)-{\mathrm{li}}(x)$ is positive" *Mathematics of Computation* **79** (2010), pp. 2395-2405.