How Close are we to Solving the Riemann Hypothesis?
“If I were to awaken after having slept a thousand years, my first question would be: has the Riemann Hypothesis been proven?” — David Hilbert
Introduction
If there is any unsolved conjecture in mathematics that has resisted attack from all sides, it must be the Riemann Hypothesis. Proposed in 1859 by Bernhard Riemann, it is now one of the seven (now six) Millennium Prize problems offered by the Clay Mathematics Institute. Any mathematician looking to become a millionaire can do so; all they need is to offer a definitive proof of the Riemann Hypothesis.
I’ve heard this be described as the hardest way to make $1,000,000.
Now, this isn’t for a want of trying. Thousands of high-level mathematicians, and an even greater number of amateur mathematicians, have attempted this problem at one point in their careers. Many people believe that they’ve found a final proof of the hypothesis only for mistakes to be found in their methods. Others have tried to tackle lesser problems in the hopes that they would provide the needed link. So far, the Riemann Hypothesis seems truly out of reach.
So, what exactly is the Riemann Hypothesis? Why are so many mathematicians studying it? And what does it mean for mathematics?
A strange function
The Riemann Hypothesis is really a statement about the Riemann Zeta Function (RZF). This function itself is ubiquitous in mathematics. It gives us a way to calculate pi, it determines how many dimensions there are in various branches of string theory, it tells us when a liquid will become a Bose-Einstein condensate, and much more (one of these unmentioned properties we’ll get to very soon).
While this function usually returns interesting values, there are some situations when it just gives us the number 0. That’s not very exciting, now is it? But what is exciting is figuring out where and when we get 0 out of this function. This function has a set of zeroes that are very easy to find, known as the trivial zeroes, and another set of zeroes that seems to come out of nowhere, the non-trivial zeroes. Oddly enough, all of the non-trivial zeroes we’ve found so far show an extremely unusual and regular pattern. The problem being, we don’t know if this pattern continues or if it will stop at some point.
This is where the Riemann hypothesis come into play: all possible non-trivial zeroes of the Riemann Zeta Function follow the same pattern. Not just the ten trillion zeroes we’ve found already, but all possible zeroes that we will find in the future. While the idea is easy enough to state, proving this statement has already shown itself to be a million-dollar problem.
But this doesn’t at all explain why people are trying so hard to prove this hypothesis. A hard problem is one thing to catch mathematicians’ interest, but a useful hard problem is outright alluring. Clearly, this function already has some very interesting properties. What the Riemann Hypothesis adds is yet another property to the list. This time, it’s about prime numbers.
Prime numbers
Let’s be honest here. Most of mathematics, in some way or another, revolves around the prime numbers and their unique properties. Several areas of active research go into finding the largest primes, progressions in primes, primes with special properties, numbers that act like primes, primes in cryptography, generalizations of the primes, primes, and primes accessories.
Perhaps the most groundbreaking achievement of 19th-century mathematics was the prime number theorem, which, when given a number n, estimates the nth prime number. In a similar fashion, it gives an equation approximating how many primes exist under a given value.
Where does the Riemann Hypothesis come into play? Interestingly enough, the Riemann Zeta Function was used in the proof of the prime number theorem. It turns out that this function can be manipulated in such a way that you get a functional expression which contains all existing prime numbers. In effect, the Riemann Zeta Function contains esoteric information about the prime numbers within itself. And if that’s the case, then it’s not much of a leap to figure out that a resolution of the Riemann Hypothesis will give us even greater insight into the primes.
But the prime number theorem did not rely on the Riemann hypothesis. At least, not in the form it was in during the 19th century. So, how will the Riemann Hypothesis help us with the primes?
As things are right now, the prime number theorem gives only approximations. Outside of that, we have no way to judge the accuracy of its output without manually verifying everything it gives us. What the Riemann Hypothesis does is upgrade the prime number theorem such that we no longer need to worry about its accuracy. It puts hard bounds on the prime number theorem and guarantees that all of its inaccuracies will remain below a fixed threshold. In fact, the Riemann Hypothesis provides the best possible threshold for this. We haven’t found any numbers that break through this threshold.
Additionally, the Riemann Hypothesis has implied other results about the prime numbers, some of which were later proven to be true. A variation of the Goldbach conjecture is one example: Every odd number is the sum of three primes. This gives us some indication that the hypothesis is true due to the fact that, so far, none of the theorems it has implied have been shown to be false.
How can it be proven?
Currently, there are several approaches to solving it. This list is comprehensive by no means, but it should give a good overview of how people are trying to approach this problem. From now on I will refer to it as RH for brevity.
Great initial progress was made by Godfrey Hardy when he proved not only that the Riemann Zeta Function has infinitely many zeroes, but also that infinitely many of those zeroes follow the same pattern conjectured by RH.
Others such as Karatsuba and Selberg have continued this line of thinking by studying how the density of these zeroes change. Roughly, they become increasingly dense at higher values. Similarly, Haddamard placed limits on where non-trivial zeroes can lie, further narrowing down their distribution. Kevin Ford later improved this limit to become more restrictive.
Some progress has been made in studying functions that are similar to the zeta function. Among them are the von Mangoldt function, the Möbius function, and Selber’s Zeta functions (which have their own analog to RH which was proven to be true).
Some attempts have been to reformulate the Riemann hypothesis into equivalent statements. That is, translating the Riemann hypothesis into another field of mathematics which may have better tools for solving it. Zagier has connected RH to ergodic theory, while Connes has converted it to a problem of operator algebra. Lagarias found an elementary mathematical statement that has a solution only if RH has one. Caveney has done something similar as well.
Others have come absurdly close to proving it, and actually have proven RH in very limited situations. One example is Andre Weil, who proved RH for finite number systems. Unfortunately, that proof cannot be carried over to the general case. At least, not yet.
An unconventional approach to it — but more of a curiosity than a serious attempt at proving it, is constructing a turing machine which halts only if RH is false. Normally, it’s impossible to know whether this machine will halt due to Turing’s Halting Problem, but this can be circumvented by knowing the “busy beaver” number for a turing machine with as many states as it has (5,372). If that seems simple enough, I should probably mention that we don’t even know the busy beaver number for a 5-state machine.
To further muddle the situation, there have been several erroneous proofs by both amateur and professional mathematicians. It seems more to be the case that we can only ever inch closer to proving RH, slightly improving every result but never making leaps of progress.
A light at the end of the tunnel
A new measure has been introduced with implications for RH: the De Bruijn-Newman constant.
Since its existence was proven in 1976, it was found that RH is true if and only if the constant is less than or equal to 0. Since then, there has been great effort at finding its lower and upper bounds. Meanwhile, Terrence Tao has been leading a collaborative research project to improve the upper bound of this constant. Currently, it is known to be below 0.22.
Very recently (and in fact the reason I’m writing this in the first place), Brad Rogers and Tao have shown that this constant is greater than or equal to 0. Thus, the Riemann hypothesis is true if and only if the De Brijin-Newman constant is 0.
What needs to be done to prove the Riemann Hypothesis is to push the upper bound of this constant down further, but it appears as if Tao’s current methods are too limited to push it all the way down to 0. Importantly, the upper bound is dependent on the highest number of known zeroes of the Riemann Zeta Function; but it’s completely infeasible, and likely impossible, to calculate enough zeroes to limit the constant enough to prove RH.
If the Riemann Hypothesis is true, then it is only barely true. We know now that only if the De Bruijn-Newman constant is 0 we will have the final proof of this 159-year old problem. Should it end up being anywhere above 0 is still a possibility. And should that happen, then a conjecture long believed to be true by the mathematics community will be invalidated.
