Conjecture25.2.1
The error in the PNT is less than \begin{equation*}\frac{1}{8\pi}\sqrt{x}\log(x)\; .\end{equation*}
Section25.2Improving the PNT¶ permalink
Let's look at a table of some of these results.
This table shows the errors in Gauss' and our new estimates for every hundred thousand up to a million. Clearly Gauss is not exact, but the other error is not always perfect either.
After the Prime Number Theorem was proved, mathematicians wanted to get a better handle on the remaining error between the log integral and \(\pi(x)\). In particular, the Swedish mathematician Helge Von Koch made a very interesting contribution in 1901.
This seems to work, broadly speaking.
Given the data in this graphic, the conjecture seems plausible, if not even improvable (though remember that \(Li\) and \(\pi\) switch places infinitely often, see Fact 21.2.3!). Of course, a conjecture is not a theorem, but luckily Von Koch had one of those as well.
Theorem25.2.2
The truth of the error estimate \begin{equation*}\left|\pi(x)-Li(x)\right|\leq \frac{1}{8\pi}\sqrt{x}\log(x)\end{equation*} for the prime number theorem is equivalent to saying that \(\zeta(s)\) equals zero precisely where Riemann thought it would be zero in 1859 (see Conjecture 25.3.2).
This may seem like an odd statement. After all, \(\zeta\) is just about reciprocals of all numbers, and can't directly measure primes. (And what do I mean by “thought it would be”?) But in fact, the original proofs of the PNT also used the \(\zeta\) function in essential ways. So Von Koch was just formalizing the exact estimate it could give us on the error.