A Statement of the Riemann Hypothesis

The Riemann Hypothesis, a longstanding unsolved conjecture in analytic number theory, is considered by many mathematicians to be one of the most important unsolved problems in theoretical mathematics. To understand the statement takes only a typical undergraduate mathematics education, but to find a proof would be the capstone of a mathematical career.

The Riemann zeta function is most simply described as a series, the infinite sum over n of 1/n^z for complex z. This series converges when the real part of z is greater than 1, and diverges when it is less than 1. The convergence properties of the series are less straightforward when Re(z) = 1. Notice, for instance, that substituting z = 1 describes the harmonic series, which diverges, so we see that the series diverges there.

Looking at the zeta function for values of z with real part less than 1 is a bit trickier. Through an analytic technique similar in flavor to integration by parts, a representation can be obtained which is good for Re(z) > 0. From there, it is possible to use a functional equation which relates values of the function with corresponding values reflected across the line Re(z) = 1/2. It turns out that the Riemann zeta function is meromorphic on the entire complex plane with a single simple pole at the point z = 1. For those without background in complex analysis, this means that the function is infinitely differentiable and has a convergent Taylor series at every point except the pole, and if you divide by (z-1), you get a function which has this property on the entire plane.

So the zeta function is, in some sense, a very nice function. In fact, there are deep connections between the zeta function and prime number theory which require some study to appreciate. Again and again, a fundamental question emerges about the function: Where on the complex plane does the function take value zero? It is possible to show that the function is strictly non-zero on the half-plane Re(z) > 1. Using the functional equation, this also shows that the function is non-zero in Re(z) < 0, except at the negative even integers z = -2, -4, -6, ... . The real question concerns zeros that occur in what is known as the "critical strip" 0 ≤ Re(z) ≤ 1. It is known that there are infinitely many of these "non-trivial" zeros, and that they have reflective symmetry across the real axis, and across the line Re(z) = 1/2.

The Riemann hypothesis claims that the non-trivial zeros of the zeta function all lie on the line Re(z) = 1/2. The conjecture, proposed by Bernhard Riemann in 1859, has been attacked by many of the most prominent modern mathematicians, but no proof has been forthcoming. However, no disproof has emerged either. Computers have shown that the first 10 trillion zeros in the critical strip, arranged in order of positive imaginary part, satisfy the hypothesis. It is commonly accepted in the mathematical community that the hypothesis is true (hence the name "hypothesis" rather than "conjecture"), but rigorous mathematicians will not be truly satisfied until a proof has been found.

Concerning Uncountable Sums

Sums are a beautiful notion which are sometimes taken for granted. 2 + 2 = 4, the sum from 1 to infinity of 1/2ⁿ is equal to 1, and so forth. In the case of countably infinite sums, questions of convergence become important. But what happens when you have a sum over uncountably many summands? It turns out that there is a simple answer to this, but it requires some care.

The typical definition for the sum of an arbitrary collection of non-negative summands is to take the supremum of sums over all finite subsets of the summands. It can be shown that if the collection S of positive summands in the sum is uncountable, then the sum must be infinite. Indeed, suppose by way of contradiction that the sum of elements in S is finite. Then the collection S_n of summands s ≥ 1/n must be finite for each n--otherwise, the supremum over all finite collections of such summands would be larger than k/n for any k. But then we may write S as the countable union over all n of the S_n, and this implies that S is countable, a contradiction with the starting hypothesis. Thus we must conclude that the sum is infinite.

Although a general sum is not defined for complex summands, the argument can be extended to this context to show that partial sums of an uncountable collection of complex numbers may have arbitrarily large magnitude. Certainly, using the same argument as above, an uncountable number of summands must have magnitude at least 1/n for some n. Further, an uncountable number of these must have argument within an interval of length at most π/4. The combination of these two conditions restricts either the imaginary or the real part of the summands, fixing the sign and providing a strictly positive lower bound on the magnitude. Since we have infinitely (uncountably) many such summands, the result follows.

A Small Subset of the Reals

Let {q_n} be an enumeration of the rationals, and for each n, let U_n be an open interval of length 1/2ⁿ centered at q_n. Denote the union of all U_n by U. Then U is a dense open subset of the reals which has measure at most 1.

From a topological point of view, this certainly seems to be a reasonable observation. After all, the rationals are dense and have measure zero, so it isn't surprising that by making the measure positive one can add the condition of openness. But from a more direct perspective, the construction seems to yield a very strange set--a spattering of droplets across the real line which cover very little length but come arbitrarily close to any point. Like an Impressionist painting, one must add a little distance to make out the comprehensible form.

Image by Claude Monet.