Section25.2Improving the PNT

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 PNT was proved, mathematicians wanted to get a better handle on the error in the PNT. In particular, the Swedish mathematician Von Koch made a very interesting contribution in 1901.

Conjecture: The error in the PNT is less than \[\frac{1}{8\pi}\sqrt{x}\ln(x)\; .\]

This seems to work, broadly speaking.

Given this data, the conjecture seems plausible, if not even improvable (though remember that \(Li\) and \(\pi\) switch places infinitely often!). Of course, a conjecture is not a theorem. He did have one, though.

Theorem25.2.1

The error estimate \[\left|\pi(x)-Li(x)\right|\leq \frac{1}{8\pi}\sqrt{x}\ln(x)\] for the prime number theorem is equivalent to saying that \(\zeta(s)\) equals zero precisely where Riemann thought it would be zero in 1859.

This may seem odd. After all, \(\zeta\) is just about reciprocals of all numbers, and can't directly measure primes. 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.