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Section 25.2 Improving the PNT

The following Table 25.2.1 shows the errors in Gauss’ and our new estimate for every hundred thousand up to a million. Clearly Gauss is not exact (recall Figure 21.2.4), but the other error is not always perfect either.
Table 25.2.1. Errors between \(\pi(x)\text{,}\) the log integral, and a Möbius estimate
\(i\) \(\pi(i)\) \(\pi(i)-Li(i)\) \(\pi(i)-\sum_{j=1}^{3} \frac{\mu(j)}{j} Li(x^{1/j})\)
\(100000\) \(9592\) \(-36.71\) \(3.882\)
\(500000\) \(41538\) \(-67.50\) \(7.087\)
\(1000000\) \(78498\) \(-129.0\) \(-31.00\)
We can build an interactive table of some results if we are online.
After the Prime Number Theorem was proved, mathematicians wanted to get a better handle on the remaining error between the log integral and \(\pi(x)\text{.}\) In particular, the Swedish mathematician Helge Von Koch 1  made a very interesting contribution in 1901.
This seems to work, broadly speaking. You can try it interactively after the static graphic.
Von Koch estimate of error in prime number theorem
Figure 25.2.3. Von Koch estimate of error in prime number theorem
Given the observed data, the conjecture seems plausible, if not even open to improvement. Though we should remember that \(Li\) and \(\pi\) switch places infinitely often, see Fact 21.2.6! Of course, a conjecture is not a theorem, but luckily Von Koch had one of those as well.
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 for the error.