On the Sign of the Difference π(x) - li(x)

A NEW BOUND FOR THE SMALLEST x WITH π(x) > li(x)

“…In the course of the 20th century, successive numerical upper bounds were found by Skewes [17], Skewes [18], Lehman [8], te Riele [15]. For Littlewood's own account of the discovery of the Skewes numbers, see [10, p. 110-112].…”

A sharp region where 𝜋(𝑥)-𝑙𝑖(𝑥) is positive

“…His theorem enabled him to show that there exists a region near 1.65 × 10 1165 where the difference π(x) − li(x) admits positive values. Then, te Riele [3] in 1987 discovered another region near 6.65 × 10 370 , and Bays and Hudson [4] exhibited a region near 1.40 × 10 316 in 1999. In 2006, Chao and Plymen [5] gave an improvement on the error terms of Lehman's theorem.…”

The prime-counting function and its analytic approximations

“…Namely, the method used by Lehman [11], te Riele [16], and finally Bays and Hudson [1] in imposing upper bounds for (see Section 1) can be used for generating rough sketches of π(x) − (x) for x -ranges far broader than those amenable to exact computation of π(x). Such sketches (see [1], p. 1291) suggest that could perhaps be in the vicinity of 10 176 , 10 179 , 10 190 , 10 260 , or 10 298 , but a smaller value seems quite unlikely.…”

“…The first unconditional upper bound for was obtained in 1955 by Skewes [15], who showed that log 10 log 10 log 10 log 10 < 3. This was strengthened in 1966 by Lehman [11] to < 1.65 × 10 1165 , in 1987 by te Riele [16] to < 6.69 × 10 370 , and in 2000 by Bays and Hudson [1] to < 1.40 × 10 316 . The first lower bound for , namely > 3000000, followed from the computations of Gauss described above.…”