whatever the precise answer to the question, it is at any rate definitely less than N. Proven upper bounds N of this sort are sometimes far larger than the actual answer.

That was the case with the first upper bound for the Littlewood violation. In 1933 Littlewood’s student Samuel Skewes showed that if the Riemann Hypothesis is true, the crossover point must come before , a number of about 10^{ten billion trillion trillion} digits. That’s not the number; that’s the number of digits in the number. (By way of contrast, the number of atoms in the cosmos is thought to have about eighty digits.) This monstrosity attained fame as “Skewes’ number,” the largest number ever to emerge naturally from a mathematical proof up to that time.⁷⁸

In 1955 Skewes improved his result, this time without assuming the truth of the Riemann Hypothesis, to a number of a mere 10^{one thousand} digits. In 1966, Sherman Lehman pulled the upper bound down to a much more manageable (or at least, writable) figure, 1.165 × 10¹¹⁶⁵ (a number, that is, of a mere 1,166 digits), and established an important general theorem about the upper bound. In 1987, using Lehman’s theorem, Herman te Riele reduced the upper bound still further, to 6.658 × 10³⁷⁰.

At the time of writing (mid-2002), the best figure is the one established by Carter Bays and Richard Hudson in 2000, also starting from Lehman’s theorem.⁷⁹ They showed that there are Littlewood violations in the vicinity of 1.39822 × 10³¹⁶ and even gave some reasons for thinking that these may be the first violations. (Bays’s and Hudson’s paper leaves open a small possibility that lower violations might exist, perhaps even as low as 10¹⁷⁶. They also show a huge zone of violation around 1.617 × 10⁹⁶⁰⁸.)

VIII. These oscillations of the error term Li(x) – π (x) from positive to negative and back take place within fairly well-defined constraints, though. If this were not so, the PNT would not be true. Some