Chapter 5

Error estimates and the Riemann hypothesis

No statement about approximation or limits is really complete without an estimate of the difference between the quantities concerned. For example, the statement that $(1 + x/n)^n$ tends to $e^x$ as $n \to \infty$ is much less informative than the statement that it lies between
$(1-x^2/n)e^x$ and $e^x$. Our other number-theoretic estimations in sections 2.5 and 2.6 all came complete with an error estimate, but so far we have no such estimate for the prime number theorem itself.

In this chapter, we will establish that for certain constants $K, \: c$,

$$\vert\pi (x) - \mbox{li}(x)\vert \leq Kx \exp [-c(\log x)^{1/2}] .$$

The ungainly expression on the right is  $o[x/(\log x)^k]$ for every $k > 0$, but not $O(x^\alpha )$ for any $\alpha < 1$. This estimate was already obtained by de la Vallée Poussin in his original proof of the prime number theorem, and despite enormous interest in the problem, it has not been seriously improved in the intervening century.

As with our original ``fundamental theorems", we will actually prove this result in the context of general Dirichlet series, and it applies equally to our second main example, $M(x)$. The method is a fairly straightforward adaptation of our original proof by Mellin inversion (though one can enter from Newman's method, at the cost of an extra step). The previous (FT3) has to be replaced by a stronger hypothesis. For the main particular cases, $\zeta'(s)/\zeta (s)$ and $1/\zeta (s)$, this means that we will need to know that $\zeta (s) \neq 0$ on a region of the form $\sigma \geq 1 - c/\log t$. The proof of this fact depends on a rather different circle of ideas, and we defer it to section 5.3, inviting the reader to take it on trust and read about the error estimate first.

A huge question mark hangs over this work. Massive numerical evidence suggests that both $\pi (x) - \mbox{li}(x)$ and $M(x)$ are really $O(x^\alpha )$ for all $\alpha > \half$, which would mean a radical improvement on de la Vallée Poussin's estimate. It turns out that both of these statements are actually equivalent to the Riemann hypothesis. In other words, the Riemann hypothesis is equivalent to the error term in the prime number theorem being what computation suggests, rather than what we have been able to prove. The importance attached to the hypothesis is a measure of the respect accorded to the prime number theorem in the world of mathematics! We go part way to explaining the equivalence in section 5.2.