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 $\pi(x)>{\mathrm{li}}(x)$](http://www.ams.org/journals/mcom/2000-69-231/S0025-5718-99-01104-7/)" *Mathematics of Computation* **69** (2000), pp. 1285–1296.


[2] T. Kotnik. "[The prime-counting function and its analytic approximations](http://lbk.fe.uni-lj.si/pdfs/aicm2008.pdf)", *Advances in Computational Mathematics* **29** (2008), pp. 55-70.


[2] R. Sherman Lehman. "[On the difference $\pi(x)-{\mathrm{li}}(x)$](http://matwbn.icm.edu.pl/ksiazki/aa/aa11/aa11132.pdf)" *Acta Arithmetica* **11** (1965), pp. 397–410.


[3] H. J. J. te Riele. "[On the sign of the difference $\pi(x)-{\mathrm{li}}(x)$](http://www.ams.org/mcom/1987-48-177/S0025-5718-1987-0866118-6/S0025-5718-1987-0866118-6.pdf)" *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](http://www.ams.org/journals/mcom/2010-79-272/S0025-5718-10-02351-3/home.html)" *Mathematics of Computation* **79** (2010), pp. 2395-2405.



Analytic lower bounds on the first sign change of pi(x) - li(x)?
There are some explicit results in the work S. B. Stechkin, A. Yu. Popov, “The asymptotic distribution of prime numbers on the average”, Uspekhi Mat. Nauk, 51:6(312) (1996), 21–88 


