The Riemann Hypothesis; the most famous unsolved problem in mathematics

I’ve read 3 books on this topic: The Music of the Primes by Marcus du Sautoy, Prime Obsession by John Derbyshire and Stalking the Riemann Hypothesis by Dan Rockmore (and I originally read them in that order). They are all worthy of recommendation, but only John Derbyshire makes a truly valiant attempt to explain the mathematics behind the ‘Hypothesis’ (for laypeople) so it’s his book that I studied most closely.

Now it’s impossible for me to provide an explanation for 2 reasons: one, I’m not mathematically equipped to do it; and two, this is a blog and not a book. So my intention is to try and instill some of the wonder that Riemann’s extraordinary gravity-defying intuitive leap passes onto those who can faintly grasp its mathematical ramifications (like myself).

In 1859 (the same year that Darwin published The Origin of the Species), a young Bernhard Riemann (aged 32) presented a paper to the Berlin Academy as part of his acceptance as a ‘corresponding member’, titled “On the Number of Prime Numbers Less Than a Given Quantity”. The paper contains a formula that provides a definitive number called π (not to be confused with pi, the well-known transcendental number). In fact, I noticed that Derbyshire uses π(x) as a function in an attempt to make a distinction. As Derbyshire points out, it’s a demonstration of the limitations arising from the use of the Greek alphabet to provide mathematical symbols – they double-up. So π(x) is the number of primes to be found below any positive Real number. Real numbers include rational numbers, irrationals and transcendental numbers, as well as integers. The formula is complex and its explication requires a convoluted journey into the realm of complex algebra, logarithms and calculus.

Eratosthenes was one of the librarians at the famous Alexandria Library, around 230 BC and roughly 70 years after Euclid. He famously measured the circumference of the Earth to within 2% of its current figure (see Wikipedia) using the sun and some basic geometry. But he also came up with the first recorded method for finding primes known as Eratosthenes’ Sieve. It’s so simple that it’s obvious once explained: leaving the number 1, take the first natural number (or integer) which is 2, then delete all numbers that are multiples of 2, which are all the even numbers. Then take the next number, 3, and delete all its multiples. The next number left standing is 5, and one just repeats the process over whatever group of numbers one is examining (like 100, for example) until you are left with all the primes less than 100. With truly gigantic numbers there are other methods, especially now we have computers that can grind out algorithms, but Eratosthenes demonstrates that scholars were fascinated by primes even in antiquity.

Euclid famously came up with a simple proof to show that there are an infinite number of primes, which, on the surface, seems a remarkable feat, considering it’s impossible to count to infinity. But it’s so simple that Stephen Fry was even able to explain it on his TV programme, QI. Assume you have found the biggest prime, then take all the primes up to and including that prime and multiply them all together. Then add 1. Obviously none of the primes you know can be factors of this number as they would all give a remainder of 1. Therefore the number is either a prime or can be factored by a prime that is higher than the ones you already know. Either way, there will always be a higher prime, no matter which one you select, so there must be an infinite number of primes.

The thing about primes, that has fascinated mathematicians for eons, is that there appears to be no rhyme or reason to their distribution, except they get thinner - further apart as one goes to higher numbers. But even this is not strictly correct because there appears to be an infinite number of twin primes, 2 primes separated by a non-prime (which must be even for obvious reasons).

Back to Riemann’s paper and its 150 year old legacy. Entailed in his formula is a formulation of the Zeta function. Richard Elwes provides a relatively succinct exposition in his encyclopaedic MATHS 1001, and I’m not even going to attempt to write it down here.  The point is that the Zeta function gives complex roots to infinity. Most people know what a quadratic root is from high school maths. If you take the graph of a parabola like y = ax² + bx + c, then it crosses the x axis where y = 0. It can cross the x axis in 2 places, or just touch it in 1 place or not cross it at all. The values of x that gives us a 0 value of y are called the roots of the equation. As a polynomial goes up in degree so does its number of roots. So a quadratic equation gives us 2 roots maximum but a polynomial with degree 3 (includes x³) will give us 3 roots and so on. Going back to the parabola, in the case where we don’t get any roots at all, it’s because we are trying to find square roots of negative numbers. However, if we use i (√-1), we get complex roots in the form of a + ib. (For a basic explanation see my Apr.12 post on imaginary numbers.) A trigonometric equation like sinθ can give us an infinite number of zeros and so can the Zeta function.

If you didn’t follow that, don’t worry, the important point is that Riemann’s Hypothesis says that all the complex zeros of the Zeta function (to infinity) have Real part ½. So they are all of the form ½ + ib. Riemann wasn’t able to provide a proof for this and neither have the best mathematical minds since. The critical point is that if his Hypothesis is correct then so is his formula for finding an exact number of primes to any given number.

In the 150 years since, Riemann’s Hypothesis has found its way into many fields of mathematics, including Hermitian matrices, which has implications for quantum mechanics. The Zeta function is a formidable mathematical beast to the uninitiated, and its relationship to the distribution of the primes was first intimated by Euler. Riemann’s genius was to introduce complex numbers, then make the convoluted mental journey to demonstrate their pivotal role in providing an exact result. Even then, his fundamental conjecture was effectively based on a hunch. At the time he presented his paper, he had only calculated the first 3 non-trivial zeros (non-trivial means complex in this context) and computers have calculated them in the trillions since, yet we still have no proof. It’s known that they become chaotic at extremely high numbers (beyond the number of atoms in the universe) so it’s by no means certain that Reimann’s hypothesis is correct.

It would be a huge disappointment to most mathematicians if either a proof was found to falsify it or an exception was found through brute computation. Riemann gave us a formula that gives us an accurate count of the primes (Derbyshire gives a worked example up to 1 million) that’s dependent on the Hypothesis being correct to specified values. It’s hard to imagine that this formula suddenly fails at some extremely high number that’s currently beyond our ken, yet it can’t be ruled out.

Marcus du Sautoy, in The Music of the Primes, contemplates the Riemann hypothesis in the context of Godel’s Incompleteness Theorem, which is germane to the entire edifice of mathematics. The primes have a history of providing hard-to-prove conjectures. Along with Riemann’s hypothesis, there is the twin prime conjecture I mentioned earlier and Goldbach’s conjecture, which states that every even number greater than 2 is the sum of 2 primes. These conjectures are also practical demonstrations of Turing’s halting problem concerning computers. If they are correct, a computer algorithm set to finding them may never stop, yet we can’t determine in advance whether it will or not, otherwise we’d know in advance if it was true or not.

As du Sautoy points out, a corollary to Godel’s theorem is that there are limits to the proofs from any axioms we know at any time. In essence, there may be mathematical truths that the axioms cannot cover. The solution is to expand the axioms. In other words, we need to expand the foundations of our mathematics to extend our knowledge at its stratospheric limits. Du Sautoy speculates that the Riemann Hypothesis, along with these other examples, may be Godel’s Incompleteness Theorem in action.

Exploring the Reimann Hypothesis, even at the rudimentary level that I can manage, reinforces my philosophical Platonist view of mathematics. These truths exist independently of our investigations. There are an infinity of these Zeta zeros (we know that much) the same as there are an infinity of primes, which means there will always exist mathematical entities that we can’t possibly know. But aside from that obvious fact, the relationship that exists between apparently obscure objects like Zeta zeros and the distribution of prime numbers is a wonder. Godel’s Theorem implies that no matter how much we learn, there will always be mathematical wonders beyond our ken.

Addendum: This is a reasonably easy-to-follow description of Riemann's famous Zeta function, plus lots more.