The Riemann hypothesis: Error estimates for the prime number theorem

The Riemann hypothesis has been just about the most notorious unsolved problem in mathematics since Riemann’s work became widely known, so it’s been researched intensively from many angles. For example, various equivalent formulations have been developed. The proof of any of them would confirm the original hypothesis. As noted earlier, even stronger hypotheses have been considered, such as the Mertens’ conjecture, which would imply the Riemann hypothesis but aren’t implied by it. (The Mertens’ conjecture was finally proven to be false.) Likewise, other conjectures have been made which are weaker than the Riemann hypothesis. They are implied by it, but don’t imply it by themselves. Up until 1896 the prime number theorem was in this category.

Let’s review a bit. The first clue that the Riemann zeta function is related to prime numbers came from the Euler product formula, which exhibits ζ(s) explicitly in terms of an infinite product (that converges for Re(s) > 1) which has factors involving the prime numbers in a simple way. From this formula it is possible to derive fairly straightforwardly many identites that relate ζ(s) to a variety of arithmetic functions. In fact, it can be shown that

log ζ(s) = s ∫2≤x<∞ π(x)/(x(xs-1)) dx

which expresses log ζ(s) explicitly in terms of an integral that involves the prime number function π(x). With considerably more effort, using ζ(s) as a key tool, Riemann was able to come up with an “explicit” formula for π(x) itself.

When the prime number theorem was eventually proved by Hadamard and de la Vallée Poussin the plot thickened considerably. Not only was ζ(s) related to the prime numbers in a formal way, but its properties turned out to play a governing role in the asymptotic behavior of π(x). More specifically, it was critical for the proof of the prime number theorem that ζ(s) should not have any zeros with Re(s) = 1.

But the connection between the zeros of ζ(s) and the asymptotic behavior of π(x) turned out to be far stronger than that. Using the alternative product formula for ζ(s) that Riemann conjectured and Hadamard proved, it could be shown that the precise distribution of the zeros of ζ(s) affected the goodness of the approximation of π(x) by Li(x). And in the best possible case, if all zeros of ζ(s) lie on the line Re(s) = 1/2 as Riemann conjectured, then the error term, i. e. the difference π(x) – Li(x), is as small as possible. Not only that, but the converse is also true: if the error term has the conjectured smallest possible form, then the Riemann hypothesis is true.

So the main conjecture which is equivalent to the Riemann hypothesis is formulated in terms of the absolute error in the approximation of π(x) by Li(x). Specifically it is the conjecture that we can write

π(x) = Li(x) + E(x), where |E(x)| ≤ Cx1/2 log(x)

It is known that this estimate is the best possible, because the difference between π(x) and Li(x) can actually be that large. What this means is that the distribution of prime numbers is so closely related to the distribution of the zeros of ζ(s) that the “best” possible estimate of π(x) can be made if, and only if, all zeros lie on the line Re(s) = 1/2.

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