The Riemann Hypothesis, Explained

When you look at this, what do you see? To me,  this is one of the most beautiful, fascinating and frustrating things i've ever seen. That's because I'm a mathematician, and these hypnotizing curly cues have haunted people like me for 150 years.

What we're looking at is a glimpse of the Riemann hypothesis. It's one of the most important  -- if not THE most important -- unsolved problems in all of mathematics. You may have heard about the Riemann hypothesis, because it's one of the millennium problems of the Clay Institute.

What that means is that the person who solves it not only achieves immortality in mathematics, but also wins a 1 million prize. But there are other reasons to care about the enigma that is the Riemann hypothesis. If the hypothesis is true, it would solve a mystery as old as math itself: That is, the mystery of the prime numbers.

Countless theorems in fields as far apart as  cryptography and quantum physics assume that the Riemann hypothesis is true, which means that we have a lot riding on this single, unproven statement. So, what is it? What is the Riemann hypothesis?

Why has the search for a proof become something of a holy grail for mathematicians? And what do these strange curly cues have to do with all of this? My name is Alex Kontorovich, and I'm a professor of mathematics.

I'll serve as your guide as we explore the Riemann Hypothesis. In order to build a somewhat sophisticated understanding of this hypothesis, what I'm going to try to do is mention some slightly advanced concepts. Now, you don't need an advanced degree  in mathematics to go on this journey with me.

What I'm going to do is try to lay the groundwork,  step by step, in a visual and intuitive way. So to begin, let's venture into the chaotic  and elusive universe of prime numbers. This is factor city, where the primes are  literally the building blocks of all numbers.

Now, every whole number is a building, and it's  been constructed out of its primes. The one-story houses are exactly the primes. Now thanks to the ancient Greek, Euclid of Alexandria, we know that the one-story buildings keep going on forever -- if you can walk that far.

But, depending on where you're standing, you might want to know: How many one story buildings do you see around you? This is what we want to understand about the prime  numbers, and what we currently don't. You see, the prime numbers seem to just pop up randomly among the natural numbers, all the way out to infinity.

In the late 1700s, mathematicians were  really starting to wonder whether it was possible to predict the places where primes would  appear. Now, a teenage boy named Carl Friedrich Gauss was especially obsessed with taming the  primes. He calculated massive tables of primes, going all the way up to 3 million, and looked for  patterns.

We can see the data that Gauss collected on the prime numbers by using something we'll call the Prime Counting Function. The graph of that function will show us exactly  where the prime numbers appear as the numbers get bigger and bigger. So, the graph stays flat until you hit a prime, and at that point it jumps up by one.

Here's what the graph looks like out to 10 -- and out to 100 -- and now out to 3 million, as far as Gauss went. At this scale, those bumps we saw early on have smoothed out, leaving a pretty pleasant curve. When Gauss looked at this data, he asked himself: Is there some other function that I know that has pretty much the same graph?

Now, as luck would have it, at around the same time Gauss was also playing around with logarithm functions. You might remember that a logarithm is  a function that undoes exponentiation, much in the same way as division undoes  multiplication. What Gauss noticed was that the graph of something called the logarithmic integral function (that's a function whose slope is 1 over log x) looked suspiciously similar to the graph of the Prime Counting Function.

Here's what they look like together. You can see why Gauss made the connection. At this scale, they're so close you can't even distinguish  them.

To understand what the slope of one over log x has to do with prime numbers, let's pay another visit to Factor City. What Gauss imagined was that if you're standing at address x, and you look left and you look right, the proportion of one story buildings, that is primes, that  you would see around you is roughly the same as one over log x. This suspicion became known as Gauss's Conjecture.

After Gauss formulated this conjecture, his student Bernard Riemann went on to  do something absolutely incredible with it. But to understand what Riemann did, we first have to go  back in time a little bit to meet someone else. This is Leonard Euler one of the world's first  students of calculus.

Later on in his life, he wrote many influential textbooks and standardized the modern math notation that we still use to this day. If you've ever written something in standard  function notation -- you know, like y equals f x -- well you can thank, or perhaps blame, Leonard Euler. In Euler's time, mathematicians were trying to understand the general structure of infinite  series.

What that means is you're trying to add up an infinite sequence of numbers, like 1, a half, a quarter, an eighth, and so on, forever, sometimes that total sum is a finite number, which  we then call a limit. If a series has a limit we would say that the series itself is convergent. In this series, our limit is 2.

On the other hand, if we were to add, let's say, one plus one plus one plus one plus one forever, well that sum would eventually become larger than any finite number, and so we would say that that series is divergent. Sometimes it can be rather tricky to  tell if a given series converges or not. So for example, this series, which  is called the harmonic series, turns out actually diverges, even though the  terms themselves are getting smaller and smaller.

This is really rather not obvious, and takes  some mathematical gymnastics to verify. Now, what about this series, the sum of 1 over the  squares of all the positive integers? It wasn't hard for mathematicians to show that this series  does indeed converge.

But the question was, what is the exact limit? Euler shot to mathematical superstardom when he saw this problem. What he realized and proved is that the limit is  pi squared over six.

Isn't that weird? What in the world does pi have to do with the squares? Well, that's another story for another time.

So, full of adrenaline from his early victory,  Euler kept going. He found the limits of the sum of one over the positive integers to the fourth power, and to the sixth power, and to the eighth power. Now, let's try to study all these series at once.

Let's write "s" for that exponent. When we do this, we define what's called the Zeta function. In other words, Zeta of two is the sum of one over the squares, and Zeta four is the sum of one over the fourth powers, and so on.

Euler proved that all of these converge, but  only when s is greater than one. But here's where things got really interesting. Euler discovered that the Zeta function could be expressed as an infinite product one -- for each prime number -- of an infinite series.

Here's what you do: For each prime number, you add up one over all the powers of that prime to the s. If you multiply all those series over all the prime numbers, you get the Zeta function. Euler's discovery hinted at an intimate connection between the Zeta function and prime numbers.

But it would take another hundred years for the full meaning to reveal itself. And that's when our friend Bernhard Riemann would  make a major breakthrough in prime number theory. And that breakthrough would potentially allow us, for the first time, to understand the profound mysteries of  prime numbers.

So, Riemann. He's born in 1826 in modern-day Germany. He's an absolutely spectacular and innovative prolific mathematician, and, among other things, his novel approach to  geometry actually lays the mathematical groundwork for Einstein's theory of relativity.

Riemann was also one of the founders of complex analysis, which is a branch of math that studies functions  with complex inputs and outputs. And he realized he might have something new to say if he studied, from that point of view, Euler's Zeta function. You might remember that the square of any  real number is positive: So 2 squared is 4, but negative 2 squared is also four.

So there is no number whose square is negative one. It turns out that it's really rather useful for  mathematicians to have a number whose square is negative one. So what did we do?

We just made one up. We called it "I" for imaginary, so whenever you see i squared what that means is  you should replace it with negative 1. This means that any real multiple of i, like 3i or 2i, these  are all imaginary cousins of the real numbers.

When you take combinations of real numbers and  imaginary numbers, you get something we call a complex number. Complex numbers might look  like two numbers, but they're actually just one number expressed as two parts: The real part  and the imaginary. The thing is, these imaginary and complex numbers they aren't imaginary at all -- they're just the natural extension of our usual number system from one dimension to two.

They're plotted in what we call the complex plane, with the real part going left to right, and the imaginary part up to down. Once you have these complex numbers, you can  study complex functions -- that is, functions with inputs and outputs living in the complex plane. And so, armed with the power of complex analysis, Riemann opened up a whole new perspective on  the world of mathematics.

Riemann wondered: What would happen if you allowed the Zeta function to take complex inputs? That is, he allowed s to be a complex number. So let's take the complex exponent s equals two plus 3i, for example.

We can plug it into Zeta and the function will look like this. Now, if we add each term, one at a time in the complex plane, we can see how the series begins to spiral beautifully. So Riemann had successfully accomplished what he  set out to do: He extended the Zeta function to the complex plane.

But just like Euler, Riemann found that the function only converged when the real part of s was greater than one. Now Riemann had a brilliant idea: He realized it was possible to extend the Zeta function to the rest of the complex plane. Riemann used a technique called analytic continuation, which allowed him to break open the hidden potential of the Zeta function.

Now, analytic continuation is an advanced concept  in complex analysis, but it's possible to make sense of it intuitively. You can think of it as a logical problem-solving method to extend the domain of a function. In order for Riemann to fill in the missing part of the domain in Euler's data function, he had to create a brand new function.

This function would take the same values as the Zeta function, where they both converge, but it had to make sense everywhere else on the plane, too. So the secret to analytic continuation is that there are really two functions at work at once. One is the original Zeta function, which has  limited scope.

But the other is this new function, the Riemann Zeta function, which extends  beyond the domain that Euler defined. Remember this? Now we can take a closer look at it and finally begin to understand its mysteries.

You see, when Riemann extended the domain of the  Zeta function, something kind of amazing happened. In the new territory that Riemann uncovered, suddenly the Zeta function can be seen crossing through the origin. What that means is for some inputs, the Zeta function evaluates to zero.

We call these places the "Zetas zeros". Some of the Zeta zeros are easy to explain. When you input a negative even integer, the Zeta function  equals zero.

But we don't need to worry about these so-called "trivial zeros" -- it's the non-trivial zeros that we need to talk about. These other Zeta zeros exhibit a very compelling  pattern, and that pattern is the central theme of the Riemann hypothesis. All of the non-trivial zeros lie inside a single region called the critical strip.

This is where the real part of s is between 0 and 1. Riemann proved that there are infinitely many zeros to be found in this critical strip. But here is the most important takeaway from Riemann's groundbreaking 1859 paper: Riemann hypothesized that all of the non-trivial zeros will lie not just somewhere in the strip, but on a single vertical line, smack dab in the middle.

We call this the "critical line" which  is where the real part of s is exactly one half. This is exactly the hypothesis that now bears  Riemann's name and the million dollar bounty. Now at this point, you might be wondering:  Why does the location of these non-trivial zeros matter, and what does all of this have  to do with prime numbers?

I don't blame you. Let me show you exactly why Riemann's hypothesis  is of such profound consequence to number theory. And let me do so by sharing Riemann's final discovery.

Remember Gauss's prime counting function, from the  very beginning of the video? Well, for technical reasons we're going to slightly modify it right now, but don't worry. We'll preserve the essence of the function.

instead of stepping up by 1 every  time we see a new prime number, which we'll call p, let us step up by log p. And we'll also do the same when we encounter p squared, and p cubed, and p to the fourth, and so on. So let me try to explain why we're doing all of this.

Riemann found a surprising connection between  this modification of Gauss's conjecture and his new Zeta function. He discovered a wave that  corresponds to the Zeta function at s equals one, and showed that Gauss's conjecture would follow, if only you could show that this wave was the fundamental frequency, which means that it more and more closely approximates the prime counting function as they both grow to infinity. So here's what the prime counting function looks like with that wave that Riemann discovered.

Riemann realized that the Zeta zeros were exactly what he needed to adjust this straight line to match the jagged counting function. Every time he added a Zeta zero, it contributed a harmonic. Here's what the first harmonic looks like when we add it on.

It ever so gently tries to smooth out the wiggles of the modified prime counting function. What happens when we plug in the second Zeta zero? Little better, right?

What about the first 10, or the first 30, or the first 60? You get the point. Riemann was able to rigorously prove that if  you add up all the harmonics of the Zeta zeros, all infinitely many of them, you get a perfect match to Gauss's modified prime counting function.

So, Riemann's hypothesis showed that the  distribution of prime numbers can in fact be predicted. The location of prime numbers is profoundly connected to the location of these non-trivial Zeta zeros. This means that if the Riemann hypothesis is indeed true, it would tell us everything we could possibly have a right to know about the distribution of prime numbers.

Needless to say, Riemann was unable to prove the full hypothesis. In other words, neither he nor anyone since has been able to show that every single non-trivial Zeta zero lies on that critical line. At one point, a massive computing project checked over one billion non-trivial Zeta zeros every single day.

The computer was looking for a single errant zero. If it found just one, then the Riemann hypothesis would fall apart. In the end, of the 10 trillion zeros that the computer checked, every single one was on the critical line.

But if you see a pattern holding again and  again and again, it's still impossible for machines to check it all the way to infinity. So brute force computation will never resolve the Riemann hypothesis. That means there's still only one way to be sure of it, and it's the same way that the ancient Greeks did  their math: rigorous absolute mathematical proof.