The Riemann Hypothesis

For Those Enamoured And Those Yet To Be Enamoured

In the year 2000, the Clay Institute, a private, non-profit foundation dedicated to the amelioration of mathematics, listed 7 problems in mathematics it felt are the most important and would further mathematics the most, to celebrate the new millennium.Of these Millennium Problems, only one, the Poincare conjecture, has thus been solved. It must be noted here that solving these problems does not mean proving the hypotheses projected to be true, a counter-example or a disproof would be sufficient to garner the $ 1,000,000 prize and indelible glory that come with the solution. Interestingly, the solver of this problem, Grigori Perelman, who, after his innovative proof of the conjecture also won the coveted Field’s Medal, often touted as the Nobel Prize of Mathematics, refused his prize money and abhorred celebrity.

The Traditional Gateway

However, we are not going to be discussing any of that today. Rather, I’d like to indulge in a little self-gratification by talking about another Millennium Problem, the Riemann Hypothesis. To all maths nerds out there, this is my love letter to you.

The Riemann Hypothesis is as insidious as it is difficult to understand. To even appreciate a small part of it, one must go back to the beginning by analysing the Riemann Zeta function, as shown below, in one of its forms.

riemann1
Figure 1: The Primordial Zeta Function, the traditional gateway into Riemann’s mind 

As any O-Level mathematics student can tell you, the expression above is simply a sum to infinity of the reciprocals of the natural numbers, each to the power s. A further analysis can be determined by realising that when s is larger than 1, the sum converges to a number. This means that as the sum is evaluated for more and more terms (larger and larger values of n) its value gets closer and closer to a certain number. For example, when s is equal to 2, the denominator of each term would be squared such that the value of each term would be much smaller than the previous one. As smaller and smaller terms are summed, it is evident that their value will approach a certain number as the value of each subsequent term would be insufficient to increase the value of the sum beyond this number.

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Figure 2: Also known as the Basel problem, the proof for this sum was found by the great Leonhard Euler

The opposite of this is a diverging sum, which simply means that the value of the sum gets larger and larger (in either positive or negative direction) for increasing values of n. The sum would make no sense to be of any value due to this. A simple example of this would be when s equals zero. The function would become an infinite summation of 1s. And an infinite summation of 1s is will shoot off to infinity and never converge to a particular value.

At this point, let us also consider the largest value of s, positive infinity (which isn’t a value but let’s pretend it is). Every value after 1 would be zero, due to their denominators being infinitely large. Hence, the smallest value of the zeta function for s values higher than 1 would be 1.

An Interesting Aside

A special case can be observed for the s value of 1, where the sum takes the form that you can observe below. Notice that the top sum (the one we are interested in) will always be larger in value than the bottom sum, due to each term in the second sum being smaller than or equal to the term in the top sum in the corresponding place.

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Considering only the second sum now, we can further simplify it to find out its value more clearly, as shown below.

riemann4

Notice that we have arrived at the same zeta function as we analysed when s was equal to zero (simply couple the halves and it will be an infinite series of 1s). Notice now then that the sum is divergent (the value tends to infinity as n goes to infinity) and because the zeta function we are interested in is larger than this sum, it will be divergent too. Even if the subsequent terms diminish in values, the infinite sum refuses to converge to a value. Yes, maths can be belligerent.

Mathematical Overload

But, back to the task at hand. When the eminent mathematician Bernhard Riemann got his hands on this function, he brought it forth into the complex plane (with real and imaginary numbers) and showed in his seminal paper how to extend the function to include more values than those for only when s is more than 1 through a process called ‘analytic continuation’. This is just fancy talk for transforming a function to another function that has more values than the original but all of the original’s values as well. It can be thought of as the mother function from which the original function broke off. The transformed function is shown below.

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Figure 3: Too young, too dumb to realise…

This super-complex function (that weird r-shaped symbol is the gamma function, a special function for the complex plane I don’t understand) still wasn’t perfect but it did allow mathematicians to calculate values of the zeta function for all values of s (which now was in complex form) EXCEPT for when s was equal to 1. However, nothing could be done about this singular point, so mathematicians termed it as a singularity, an oddity that did not conform to the studied function and stayed belligerent. Maths can also be petty it seems.

Let’s put in some pictures now. Below is the graphical image of the analytically continued Riemann zeta function. For all values except 1, there is now a value for the zeta function that is particular (not infinity, an actual numerical value). These values include zeroes. The O-Level mathematician now must be intrigued as the zeroes of a function are called the function’s solutions. So too, it would seem, was Riemann. After studying these zeroes, Riemann noticed two things. Firstly, all values of s which are even and negative that have no imaginary part (s = -2, -4, -6…) would equal zero due to a peculiarity in the function. This made looking for them too easy and obvious; hence they were termed trivial zeroes, zeroes which behaved in an understood manner.

Reminding ourselves that we had already seen for all s values higher than 1, the function tends to a value which will always be greater than zero. This is still true regardless of the complex nature of the sum in question since its real part will always be a sum of the real parts of the terms (values always greater than zero). With that being said, it is now clear that if any other zeroes exist, and an infinitely many of them do, they must lie in the critical strip, when the real part of s values range from 0 to 1.

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Figure 4: Zero-strip or Zero-line? That is the critical question

Riemann went further by commenting that every zero of the zeta function that is non-trivial must lie on the critical line of Re(s)= ½ (which means the real part of s must equal ½ while the imaginary part of s can be whatever value).  And so we have come to the Riemann Hypothesis. From Riemann’s time around the mid-1800s until now, the Riemann hypothesis has neither been proven nor disproven. In fact, the first 10,000,000,000,000 non-trivial zeroes have been found to lie on this critical line. So, it must be true, mustn’t it?

This is the peculiarity of mathematics. It is rigorous, and proofs are paramount. Without a formal proof the Riemann hypothesis will always be just that, a hypothesis. An assumption based on, first, intuition then a huge set of examples. But that has not deterred mathematicians from basing a large deal of their work on it, forming mathematical logic and solutions even if they might be wrong (a hugely unlikely phenomena and, even if true, would not negate the examples presented in favour of it).

But why does it matter? Why do so many people revere the Riemann hypothesis, so much so that even Andrew Wiles, the famous mathematician who solved Fermat’s Last Theorem, referred to it as the greatest problem for mathematicians now? Probably the most concerned area of mathematics with the Riemann hypothesis is prime number theory. To put it in simplistic terms, the nature of prime numbers and their properties which have been expanded upon in many theorems regarding them rely on the Riemann hypothesis being true. The very foundation of mathematics, the atoms that make up other numbers, the primes will be completely mysterious again with the negation of the Riemann hypothesis. And this effect will ripple through to practical things like cell phone technology which rely on the prime number theorem to function properly.

As a final round up of this article, I’d like to reveal my intentions behind it. I’ve always had a deep admiration for mathematics. I’ve never been any good at it but would always marvel at its beauty and be quieted by its resolve to be exact. In that regard, I think one of the most important aspects of solving the problem of the Riemann hypothesis is to uphold the philosophy of mathematics, pursuing it for maths’ own sake. With each discovery building on the last, with logic strengthening logic onwards to a dream of understanding. And so it would seem that maths can also be beautiful.

P.S If you would like to learn more about casual mathematics, do go over to the YouTube channel Numberphile. There you’ll find a plethora of videos regarding difficult concepts in Mathematics, explained such that anyone can understand. Particularly check out The Key to the Riemann Hypothesis (https://www.youtube.com/watch?v=VTveQ1ndH1c) to find out how you, yes, you, can try to prove or disprove it yourself.

P.P.S Also head over to the Clay Mathematical Institute to find out more about the 5 other Millennium Problems. It has been a pleasure this week, see you Wednesday!