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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.

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Just to be clear: is the lower limit of your definition of $\mathrm{li}$ 0 (i.e., Cauchy principal value integral) or 2? – J. M. Dec 7 '10 at 14:00
@J. M.: I was taking the Cauchy principal value where li(2) = 1.045..., but I would be interested in an answer to either, provided of course that x >= 8 to avoid small answers. – Charles Dec 7 '10 at 15:11
up vote 4 down vote accepted

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

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Thank you. I found the translation S. B. Stechkin and A. Yu Popov, Russ. Math. Surv. 51 (1996), pp. 1025-1092. – Charles Nov 12 '12 at 15:54

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