The Prime Number Theorem

Prime numbers have fascinated mathematicians since they were mathematicians to be fascinated, and the prime number theorem is one of the crowning achievements of the 19th century. The theorem answers in a precise form, a very basic and naive sounding question. How many prime numbers are ever proved in 1896? But there are marked the culmination of a century of mathematical progress and is also at the heart of one of the biggest unsolved problems in mathematics today.

With me to discuss the prime number theorem are Simon Meyssan, a fourth year student in mathematics at all college, to find Crist, a first year student in mathematics at Keibel College and became a first year student, also in mathematics at Baylor College. Thank you very much for joining me.

Before we begin, we should mention that we are recording this podcast as part of two Oxford podcast series, Secrets of Mathematics, and the recently started series In Our Spare Time, which has a rather more general agreement this year, is the title of the show is the Prime Number Theorem. So we better start with the basics. What is a prime number show? So we are looking at the natural numbers as we like to call them in mathematics. So that's the counting numbers. One, two, three, four and so on.

And we say that one of these is a prime number if it's divisible by only one at itself. So we're thinking two, two, three, five, seven, but not nine because nine is three times three. Exactly. Exactly. These things arise quite naturally in study of the whole numbers. Can you explain why these numbers are important when they come up? So looking at any of these natural numbers, we know that we can factor into prime numbers so we can write it as a product of numbers.

And this can be done in a unique way up to the ordering of the numbers. So if we think of an example, say the number 40, I guess that's that's two times two times two times five. And there's only one way of of doing. Yeah, sure. You can relate to the five spot, but as I said, the same number, I get the question. I mean, do they go on forever? Do we when we run out of numbers, they do indeed go away, as was proved I suppose, by Euclid, 300 B.C. or something like this.

So what he noticed was that so if you multiplied together those numbers and then you add one to this. So let's say I take two times three, two of these problems and add one. OK, so suppose we thought that the only prime numbers in the world return for sure we didn't. We thought two and three all over the world. OK, but what if I tried to. Yeah, I multiply two by three and I add one so I get seven.

Now this number can't be divisible by two and it can't be divisible by three because you'll get remainder one because of the way I constructed this. So seven can't be divisible by any of the known primes, let's say. But it must have a prime factor because you can told us that all numbers, I mean, even more than just the prime factors can be written into products of primes.

Exactly. So in this way, we've constructed now a number which must have a new prime factor, which was known all could be private, something like seven happens to be. So if we generalise this idea to not only two or three of the new five, but let's say we have lots of no primes, we multiply them all together, we add one and the same argument goes through anyway. Either our new number is a private self. What's divisible by some new private?

So if I was labouring under the impression I had, there are all the problems in the world in my box. You'd come along and you just multiply them all together and add one, and then that would force me to include a new problem in my box. So I couldn't have had a finite number to begin with. Yeah. Jamie, perhaps you could tell him we're skipping ahead two millennia now around sort of late 18th century, there was a conjecture made about how many numbers they were up to a threshold.

OK, so there's this mathematician that's quite well known amongst mathematicians called Carl Friedrich Gauss. And around the age of 15 or 16, when he was doing various experiments with numbers, he noticed that if you just did so by experiment, you don't necessarily mean, you know, in a lot of the tests you. But you mean he's doing all the calculations. He's yes. He's just trying to work out something. He's doing lots of calculations and looking for patterns.

So and so he noticed that if you take let's say you pick an actual number, let's say I'm looking at an arbitrary one called and for example, and you tick the numbers and and plus one and plus two and you go all the way up to a thousand. And you look at all the primes in that period, that group of numbers as you go as energy changes, the number of primes in that group of numbers decreases by a factor of about one over the logarithm of any.

Where the logarithmic end is a special kind of function, I guess we should take a bit of time to actually talk about what this function is, because it's going to play a major role in the precise statement of the prime number theorem. Simon Johnson, explain for us the longer term. I throw in one number, I get another number out. But what sort of properties does this function the logarithm have?

Well, there are actually many functions which are called logarithms, and perhaps the easiest one to explain is the logarithm to base 10, we say. So. This is roughly speaking, this is the number of digits it takes to write down the number, for example, the logarithm of ten is one by ten, logarithm of ten is one. The best ten logarithm of 100 is to the best of a thousand is three. And what's happening here is that the number of zeros is going up.

You're writing down, you write down the number and the best ten logarithm goes up with within that number of zeros, that number at a number of decades. Now, you may have heard that you don't have to write down numbers using tens, hundreds and thousands as we do. You can use you can use a different base instead of ten. So, for example, you can use some write down in binary, which is how computers store numbers. And this uses the number two instead of the number 10.

In this case, I got into the base, too, I guess, to take, you know, two to the power of five cents, thirty to the locker room with kind of the inverse of that processor instead of computing to talk to you two times, two times to two times, it's about five times being 32. The longer in about the inverse question is how many times did I have to multiply two by itself to get 30 to.

Yes, exactly, exactly. And the best 10 logarithm of a number is the number of times you have to multiply 10 by itself to get that number so long instead of one hundred two thousand three, as you said before. Exactly what the prime number theorem uses, a thing called the natural logarithm. So that's the logarithm to a very special base. Yes. So it's it's the logarithm to to the base e e being a special number, beloved of mathematicians.

It's not it's not a whole number. So this this idea that it's the number of digits you use, one won't quite work. You can't write down numbers using the base E instead of the base ten. It's, if you like, somewhere in between logarithm to the base two and logarithm to the base three.

It's distinguished by special properties when we come to look at calculus, when we come to try and differentiate an integrated in some sense just the same kind of object to this, look to the base 10, the base to in fact, it's related by it's a constant multiple.

So each of these different bases. But it kind of has a special place because of these characters properties, which we won't describe in too much detail and then show the derivative reacts of the natural log of X is one over X and you get a very similar relationship with these other log functions. But you're wrong by constant. And so there's this very special function, this natural rhythm function, and now we go back to Gemito.

What was Goussis observation, connecting primes and this logarithm function. So as I said before, Gauss noticed that when you sort of take these groups of one thousand consecutive numbers and let's say starting with N as N grows and changes the number of primes in this group of numbers, I say decreases by approximately one over the logarithm.

Then he kind of conjecture to put forward this idea that perhaps the number of primes less than a given number might be somehow connected to the logarithm event in a very precise kind of way. I think if I recall, he kept a diary. He was very precocious teenager and of insightful comments that people found many years later. And he had this observation that the density of primes around and C looks like it's about one over log and one over the natural look of it.

And so if you kind of integrate this up by something over all scales, you get a conjecture. But the number of primes less than N is going to be about and divided by the natural look of it. I mean, so how big a number is that? Because obviously not every number its problems or anything less than n but so this log function, it does tend to infinity with and tells very pretty slowly. Yeah. Yes. So hang on, let me think for a moment.

If I've done my mental arithmetic correctly, the the natural logarithm is roughly twice the base, 10 logarithm. Well, very roughly. Very roughly. So the National Laboratory of a million is very roughly 12. So a million it's a very large number going in, but it's logarithm is very approximately 12.

It's a smallish number. So we're suggesting that there is going to be roughly again, because we don't know precisely what we mean by this, about a million divided by 12 prime numbers, less than a complete. The girls can prove this conjecture is an observation. I think it's quite a romantic story, actually, because he was looking at these tables that he had an opposite pages. He had a table of the first thousand numbers and a long table.

And it was only because of this proximity of these two piece information that he got drawn into thinking, oh, I wonder how many. Oh, it looks like these logs on this other side of the page. We go back to Euclid's argument, it gives us some phone numbers, how many does it give us? Yes. So as you mentioned, it doesn't give you a verdict like when we started with two or three with no numbers. And the next one we got was seven. And then after that, we got 43 three.

And obviously we're keeping almost all the numbers like now, lots of prime numbers between what's in the military. So I think we should get roughly logarithmically many prime numbers in this way. So that means that, as Jamie was saying, Gousse conjecture that we should have in divided by log in, which is basic we should think of very nearly every day. I mean, so it's still a proportion turning to zero because law does tend to infinity.

But, you know, a lot of crimes out. That is the conjecture that counts for as far as Euclid's argument will give you lock in primes, which is, as Simon was describing, a very small number. So less than a million that would give you about 12 primes, which is obviously completely wrong. OK, so we have two problems here. One is the very precise conjecture of really exactly this many crimes for which even struggling to find more than logarithmically in any crimes.

But I guess maybe this is where we can start talking about. So this is and skipping ahead about 50 years from Kansas, about 1852, I think. Jamie, you're going to talk a little bit about intelligence estimates. OK, here, Chevy showed was a Russian mathematician and he didn't prove the prime number theorem, but he was able to provide quantities of bones, the or the number of primes less than or equal just in terms of the function and over log, which appears in the prime number theorem.

Actually, the bonds that he proved were that for some number B, which was which was almost one, he showed that the number of primes less than or equal to any lies between an overload and B and six and over five longer and B six over five. That's about one point two. So he got it like pretty close on both sides. But I mean, that that's still not quite enough. Two things to say here. One is that so we might say this is an order of magnitude estimate.

So he showed that the order of magnitude of the growth of the the number of times less than N is indeed of the order of magnitude of a log. But I think we should now maybe chat about patrician's methods, but actually just nail down what the precise statement of the prime number theorem is so we can tell how this is, I mean, further than what we have managed to create. So I'm want to tell us how good an estimate was. Goussis conjecture.

Well, the prime number theorem is is usually stated by mathematicians in terms of the number of primes between one and x axis, some large number. So informally, the conjecture is that this the number of primes between one and X is about X on log X, as we've been saying. And more precisely, the claim is that the difference between the number of primes up to X and X on Log X is much smaller than X on log X. So the difference, the error, if you like, in this estimate is.

Smaller than any fixed multiple of X on log X. This error, yours, it can still go to infinity, but just much slower than the maintenance of a log. Yes, if you like the the ratio of the number of primes up to X to. The quantity X on Block X will become very close to one when X is large, so that was the conjecture that was going around to the 19th century, which indeed is now the theory and the prime number theorem. Going back to Jamie, that's not quite what you have managed to prove.

What trebuchets showed was that this ratio lies between about nought point nine and one point one. It's. Pretty, pretty good, but I mean, it was and it was the first real reason, this is a major, major issue in political terms. Yeah, actually, it was the first major step towards the actual proof of the what? There were probably one of the first things I actually showed that it was open to attack. And now enter Reman Reman versus a single paper on No3.

And I think it's a fair trial to tell us a little bit about the ideas that were introduced to try to understand the this. So one of the most central tools which we introduced is something we call the Raymond Z definition, which is well, in at least the first proof of the theory we have, which is you can get to this function, plays a major role in this proof. What's the definition of the to function? Right.

The definition of the function evaluated in some point it's called s is that you sum all the natural numbers from one all the way to infinity and one divided by in to the power of S. So it s was like two or something. We could be computing one over one squared plus one of two squared plus one of the three squared plus one of the fourth. Exactly. And so that's just some number. That's Veeteren two.

Exactly. Is that number. Why is this useful, so, yeah, so oil, amid this very nice observation that you can fact rise this some of so in the case of Zeder to Waveband, one over one squared plus one of the two squared and so on, we can factorisation instead as a product over all prime numbers. Uh, so, so now we have a product of one divided by, uh, one plus one over P squared. So in a rather similar way that we can write all of them as the product of primes.

In fact, that fact allows us to prove that we can write this infinite sum of a product over primes of these simpler expressions, one over one minus one over three decades. Reginald's two. It's one of the P squared oilor talked about this well as one to three. But Reman extended this for Reman. What type of object was s so so Reman Reman considered the this Zitter function so called when s was a complex number. The complex numbers are what are called the real numbers.

All of the decimals, if you like, PI and 10.1 recurring and all these numbers that you can write out with an infinite number of tickets after the decimal point. And in addition you throw in the imaginary unit, the square root of minus one. So this is this is referred to as an imaginary number because there is no decimal number you can write down, which when you swear you get minus one, no matter what number you write down, if you square it, you get a positive number.

But if you define I to be some unspecified number with one square deal, it's minus one. You can you can form a perfectly good number system, if you like, by multiplying I by real numbers and adding it to real numbers to create things like PI plus two.

I for example, one way of thinking about the complex numbers is if you if you if by some lucky chance you should think about the real numbers on on a number line, a straight line with zero marks and one marked and every number having a place along this line, then the complex numbers live in live in a plain, flat, two dimensional space with zero marked and one marked. And then I marked ninety degrees at a right angle to one.

It's a shame that we're a radio podcast, but you can think very visually about the complex numbers, which I think is a very useful thing to do. So you can think in like two axes, you have a horizontal real axis of some U.S., your number line from school, and then you have another axis, the imaginary axis going a 90 degrees eye on the table in front of me. I'm crossing my hands at right angles. I you can't see, which is not very helpful to hear.

So you define for us the remains of the functions of infinity as one of what we ask as one or two. Yes. So. Does that define the function almost entirely? I'm afraid it doesn't. If that were the case. So what happened here was that I define it as an infinite sum and not always leave. Each of the numbers in my song is huge and I keep adding up huge numbers. I'm just going to get something which just grows and grows and grows and doesn't it doesn't make sense.

It doesn't really make sense. So you have to put some kind of condition on what numbers you put in so that this doesn't happen. So it turns out that the precise condition you get is that the real part of your number is has to be greater than one. So this is like it's horizontal coordinate. Everything of our axes has to be quite far to the right. It's got to be at least one. The number has to be to the right of the number one.

So what happens elsewhere is kind of here be dragons has been just defined as some crazy thing on some portion of the complex plate. So luckily for us, we have this concept which allows us to so we can we can think of the kind of remounted function to be defined, not to begin with, only if the horizontal part is to the right one. But then we have a theorem in mathematics which allows us to extend this function also to the left of this one.

So what happens is that my definition, writing it as a work to give you the right answer, but there's still a unique way of kind of getting a value service to the function, assigns a value to every point to the rights of of one to think of that as maybe some sort of smooth rubber sheet above the plane, which is, you know, how high it is above the plane is how big the function is that it's a slightly tenuous analogy to me. And it's a fair just mentioned this. It's a continuation, it's saying.

But that is a way I think that is a unique way of building a bit more of a sheet over to the left, to the left of one, which smoothly continues the spreadsheet. And so we can actually get a function on on the entire plane. Apart from one point, there's one point apart from S equals one, which is this point we've been mentioning all the time. So this is the actual point on the horizontal axis. This is a huge amount of abstract machinery.

OK, so Raymond didn't invent this complex numbers have been going around for about a century before we became law silent. Could you be, you know, for slightly more explicitly how we can get a handle on the crimes through these different. Well, we've said the part of what makes the remedy to function special is that it has this expression as it's an infant son, but it has an expression as a product of a price and.

There is, in fact, a way to construct from the reproductive function something which has an expression, as are some overprice or some other powers of prime precision is required. But to all intents and purposes, one can construct from the retina to function. Another complex function is another function of the complex plan, which can be expressed as a sum over, not over all the numbers, but a sum over all the prime numbers. To be precise, one one takes the logarithm of the Remzi to function.

That one takes the derivative of that function. So we've constructed this function using calculus by taking a derivative and in fact by using calculus, again, by taking an integral of this new function, one can pick out the terms involving involving primes between one and X. And one can go from this inference for all the primes to a sum over primes between one and X. It's the thing called Pearlman's formula, I guess you're thinking of.

So this is all happening around 1860, but Raymond died quite shortly afterwards in a short, tragic life. And although he made some great insights into this function, he did manage to prove that there was a key technical stumbling block to estimating this integral that would give you the number of times less than Jamie. Do you want to comment on on what this key stumbling block was?

Well, in order to estimate the integral, which appears when you take the sum of the prime numbers, less than a quarter X Reman realised that what you needed to do was shift the interval from one line to another. The process of doing that involves using a result known as coaches' there.

But to do this, the terms which appear inside the interval, the things that we are integrating into this function involving the logarithmic derivative of the inside function, needs to satisfy some particular properties inside the region. Between the two lines that we're moving around interval between that those that property can be summarised basically by saying that we would like the Remzi to function to be. Nonzero inside and that region between two lines of integration.

And really, the proof of the prime number theorem was facilitated by high demand and certified to coming along at the end of the 19th century and showing that this not zero free region existed. Reburn provided for Flatpack. If you could prove this theory free region for the Zyda function, then you could prove the point of no theorem. But it took another 35 years until 1896 to show that Zettl one plus it. So what is the real part? It is the imaginary part that was never zero.

And so, as Jamie described, you can move the integral and the integral is easier to estimate when you move it. Simon, obviously we have another general audience, but can you describe to us that the key idea behind this proof of the theory of region? Well, the first proofs that the Raven Zeta function possess this zero for region were relatively long and difficult compared to the compared to the demonstrations we find in our textbooks these days.

But the underlying idea is much the same. And that's that all one really needs to show, first of all, is that the and to function has no zeros on a line, on a certain straight line in the complex plane. And one can one can show this by comparing the values of the Roman to function near three points, one where it has this is a zero that we would like to prove doesn't exist. One is at the point one near the number one where we know that the Roman Zeta function is very large.

And the third point is, if you take one step from one to the point where there's supposed to be a zero, you take another step the same length. And that's the third point. If the Regency to function has a zero sum with real one. So if there is indeed a bad point with the logarithmic derivative of the Ravens, they did function, has a has a poll, and we need to move this integral part past the poll.

And that's going to that's going to mess up the plan. We can deduce that the Raymonde to function must, in fact, have another poll, a poll with real part one and it Macary part twice the imatinib part of the zero. And we know that this isn't so. We know that the only poll of the frequency to function is that one.

And this will this will enable us to to prove by contradiction that it's not possible for the presidency to function to have a zero real parts equal to what's in the details of the argument that Simon has just mentioned. It all comes down very curiously to the cosine double angle formula, which I think any of our listeners who did as level maths will have learnt.

And I remember thinking when I was there during my Masters that MacGraw almost view almost the prime number theorem as a corollary of the cosine double angle formula. OK, so we reached nineteen hundred and we proved the problem with him.

And the curious state of affairs was that people began to realise that this complex analysis approach that Simon Cy-Fair and Jamie have been describing was not just one particular method that they had managed to successfully use to prove the point of the theorem, but in some ways was equivalent to the prime no theorem in a very precise sense that the prime number theorem was equivalent to showing that there were no zeros or the Remzi to function on the one line with real parts equal in one.

And so the general consensus was that this complex analysis machinery was completely necessary to prove the prime number theorem because it was equivalent to it. So there could never be what physicians call an elementary proof. So there was this assumption, correct. So it turned out, rather surprisingly, that this assumption was not correct. In 1948, 49, a Norwegian mathematician named Unassembled, came along and in fact did give an elementary proof of the prime number theorem.

And by elementary, you mean Sarabhai Elementary. You mean here that there's no use of complex analysis and tools like the Raymonde to function so the function doesn't get a mention. It doesn't get mentioned a single time is proof, but secretly we know it must be behind that somewhere because this proof has to show there are no zeros on the one line at all. And how does the proof go rough? It splits into a couple of stages, right.

So I guess roughly we can say the proof splits into two parts. So the first part is something we call sailboat's form sandbanks formula, which instead of considering sums of primes so Asama over primes, we are considering products of two primes up to two primes. And it turns out rather crucially, that in this approach, the term which should come from the zeroes of the Remzi to function in the kind of original proof completely cancels.

So we don't actually need to know anything about the zeros the other and the council and that's it. And they have a second part of the proof, which is to actually show that these statements about sums of products of primes is strong enough to give the original prime number theorem. And that is a elementary but rather involved argument, if I remember correctly, right, Jamie?

There was quite a controversy at the time and so many years later between Salberg and a Hungarian mathematician called Paul Aatish, who also claimed to have found this elementary proof around the same time, Paul, politicians, a lot of mathematicians will know who is quite a prolific publisher of material.

He is well known enough that in a similar way to which we have back in numbers for actors, we also have Aatish numbers from mathematicians and he claimed that through collaboration with other outlets. So sorry. And so it's if there is a Norwegian pronunciation expert, I'll just refer to themselves and ask them that.

In collaboration with Sjoberg, he could also claim credit for the proof that Sjoberg provided of the prime number theorem using elementary methods because he claimed that without some communication or idea from him that he had given to Salberg, Salberg would not have been able to complete the second part of the proof. And there's an apocryphal story surrounding this sort of feud whereby Salberg is supposed to have overheard in some university maths department.

Another mathematician saying that Paul Aatish and some Norwegian fellow had found an elementary proof of the prime number theorem to which he strongly objected and and went ahead and published his proof of the prime number theorem under his own name.

There's an article online by by Goldfields that tries to clear up all the correspondence about who came first and what information passed between who I trust and SALBERG, which I invite you to read, but I'm not sure it completely clarifies the issue. To close, as I mentioned in my introduction, that the problem with there was at the heart of one of the big unsolved problems in mathematics today, Simon Johnson. Tell us what that problem is.

So you're talking about the dream hypothesis, which is so we've we've discussed this idea that in order to prove the prime number theorem, one needs to show that the the agreement Zitter function does not vanish at any point on a certain line in the complex plane. And the the Riemann hypothesis is a contract, a description of where all the zeroes of the agreement theta function are all the points, whether in such a function vanishes.

And the conjecture is that apart from some some very predictable, boring, well understood zeros, all the zeros lie on the on the line real party equals one half. So this is a vertical line in the complex plane, a distance one half to the left of this. This real path equals one where we prove there are no zeros and that the number theorem, the conjecture is that in fact, all the zeros lie a distance one half to the left of that line.

If we want to translate this property of the Regency to function back into some property of prime numbers, the written hypothesis were to be shown to be true would tell us the number of primes up to X would equal X or the log X, plus a very small error plus an error that was about the square root of X, a little bit larger than the square root of X, but about that size at the moment,

you would become a very famous mathematician indeed if you could show that the error was almost most X to the power. Nought point nine nine nine nine. So we're nearly at the end of the show, but do you have any closing thoughts that you'd like us to go home?

Well, I think something which I find interesting about this, this area we've spoken about the fact that there is there is an elementary proof of the prime number there and there is a proof of the prime number theorem, which although it is perhaps complicated, certainly a little difficult to understand at first I personally struggled to understand it. There is a proof which doesn't use this this machinery of complex analysis in the logarithmic derivative and these zeros in the complex plane.

The way this strikes me is that perhaps the perhaps this this this machinery of complex analysis isn't necessary to prove the prime number theorem, but it might in some ways still be the best way to understand it.

These connexions between different areas of maths in this case, between complex analysis and number theory, are often things which lead to new directions for a can have quite surprising implications for still other areas of maths and in this case, that are complex analytic formulation of the problem number theorem.

I think it's fair to say, has led to some some very far reaching generalisations, and although we certainly don't have time to go into this, I think one could argue fairly convincingly that it's connected, for example, to Andrew Wiles, proof of last there. Well, thank you very much, Sophia. Jeremy Simon, we've covered a lot of ground in 45 minutes. The next week in our spare time, we'll be talking about Shakespeare and music.