NTIC Improving the PNT

Section 25.2 Improving the PNT

The table 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.

\(i\)	\(\pi(i)\)	\(\pi(i)-Li(i)\)	\(\pi(i)-\sum_{j=1}^{3} \frac{\mu(j)}{j} Li(x^{1/j})\)
\(100000\)	\(9592\)	\(-36.71\)	\(3.882\)
\(500000\)	\(41538\)	\(-67.50\)	\(7.087\)
\(1000000\)	\(78498\)	\(-129.0\)	\(-31.00\)

Figure 25.2.1. Errors between \(\pi(x)\text{,}\) the log integral, and a Möbius estimate

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 made a very interesting contribution in 1901.

Conjecture 25.2.2.

The (absolute value of the) error in the PNT is less than

\begin{equation*} \frac{1}{8\pi}\sqrt{x}\log(x)\text{.} \end{equation*}

This seems to work, broadly speaking. You can try it interactively after the static graphic.

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.

Theorem 25.2.4.

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 25.3.7).

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.