Refinements to the prime number theorem

Let π(x) be the number of primes less than x. The simplest form of the prime number theorem says that π(x) is asymptotically equal to x/log(x), where log means natural logarithm. That is,

$\lim_{x\to\infty} \frac{\pi(x)}{x/\log(x)} = 1$

This means that in the limit as x goes to infinity, the relative error in approximating π(x) with x/log(x) goes to 0. However, there is room for improvement. The relative approximation error goes to 0 faster if we replace x/log(x) with li(x) where

$\text{li}(x) = \int_0^x \frac{dt}{\log t}$

The prime number theorem says that for large x, the error in approximating π(x) by li(x) is small relative to π(x) itself. It would appear that li(x) is not only an approximation for π(x), but it is also an upper bound. That is, it seems that li(x) > π(x). However, that’s not true for all x.

Littlewood proved in 1914 that there is some x for which π(x) > li(x). We still don’t know a specific number x for which this holds, though we know such numbers exist. The smallest such x is the definition of Skewes’ number. The number of digits in Skewes’ number is known to be between 20 and 317, and is believed to be close to the latter.

Littlewood not only proved that li(x) – π(x) is sometimes negative, he proved that it changes sign infinitely often. So naturally there is interest in estimating li(x) – π(x) for very large values of x.

A new result was published a few days ago [1] refining previous bounds to prove that

$|\pi(x) - \text{li}(x)| \leq 235 x (\log x)^{0.52} \exp(-0.8\sqrt{\log x})$

for all x > exp(2000).

When x = exp(2000), the right side is roughly 10⁸⁵⁷ and π(x) is roughly 10⁸⁶⁵, and so the relative error is roughly 10^-8. That is, the li(x) approximation to π(x) is accurate to 8 significant figures, and the accuracy increases as x gets larger.

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[1] Platt and Trudgian. The error term in the prime number theorem. Mathematics of Computation. November 16, 2020. https://doi.org/10.1090/mcom/3583