1 00:00:01,620 --> 00:00:06,720 Prime numbers have fascinated mathematicians since they were mathematicians to be fascinated, 2 00:00:06,720 --> 00:00:11,370 and the prime number theorem is one of the crowning achievements of the 19th century. 3 00:00:11,370 --> 00:00:16,380 The theorem answers in a precise form, a very basic and naive sounding question. 4 00:00:16,380 --> 00:00:20,880 How many prime numbers are ever proved in 1896? 5 00:00:20,880 --> 00:00:24,780 But there are marked the culmination of a century of mathematical progress and is 6 00:00:24,780 --> 00:00:29,910 also at the heart of one of the biggest unsolved problems in mathematics today. 7 00:00:29,910 --> 00:00:38,580 With me to discuss the prime number theorem are Simon Meyssan, a fourth year student in mathematics at all college, to find Crist, 8 00:00:38,580 --> 00:00:47,100 a first year student in mathematics at Keibel College and became a first year student, also in mathematics at Baylor College. 9 00:00:47,100 --> 00:00:48,720 Thank you very much for joining me. 10 00:00:48,720 --> 00:00:56,400 Before we begin, we should mention that we are recording this podcast as part of two Oxford podcast series, Secrets of Mathematics, 11 00:00:56,400 --> 00:01:02,370 and the recently started series In Our Spare Time, which has a rather more general agreement this year, 12 00:01:02,370 --> 00:01:06,930 is the title of the show is the Prime Number Theorem. So we better start with the basics. 13 00:01:06,930 --> 00:01:13,490 What is a prime number show? So we are looking at the natural numbers as we like to call them in mathematics. 14 00:01:13,490 --> 00:01:17,340 So that's the counting numbers. One, two, three, four and so on. 15 00:01:17,340 --> 00:01:23,820 And we say that one of these is a prime number if it's divisible by only one at itself. 16 00:01:23,820 --> 00:01:30,280 So we're thinking two, two, three, five, seven, but not nine because nine is three times three. 17 00:01:30,280 --> 00:01:36,780 Exactly. Exactly. These things arise quite naturally in study of the whole numbers. 18 00:01:36,780 --> 00:01:39,990 Can you explain why these numbers are important when they come up? 19 00:01:39,990 --> 00:01:49,770 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. 20 00:01:49,770 --> 00:01:54,160 And this can be done in a unique way up to the ordering of the numbers. 21 00:01:54,160 --> 00:02:01,530 So if we think of an example, say the number 40, I guess that's that's two times two times two times five. 22 00:02:01,530 --> 00:02:09,480 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. 23 00:02:09,480 --> 00:02:10,540 I mean, do they go on forever? 24 00:02:10,540 --> 00:02:21,150 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. 25 00:02:21,150 --> 00:02:29,550 So what he noticed was that so if you multiplied together those numbers and then you add one to this. 26 00:02:29,550 --> 00:02:33,750 So let's say I take two times three, two of these problems and add one. 27 00:02:33,750 --> 00:02:40,830 OK, so suppose we thought that the only prime numbers in the world return for sure we didn't. 28 00:02:40,830 --> 00:02:44,940 We thought two and three all over the world. OK, but what if I tried to. 29 00:02:44,940 --> 00:02:49,410 Yeah, I multiply two by three and I add one so I get seven. 30 00:02:49,410 --> 00:02:57,570 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. 31 00:02:57,570 --> 00:03:02,580 So seven can't be divisible by any of the known primes, let's say. 32 00:03:02,580 --> 00:03:07,050 But it must have a prime factor because you can told us that all numbers, I mean, 33 00:03:07,050 --> 00:03:10,920 even more than just the prime factors can be written into products of primes. 34 00:03:10,920 --> 00:03:15,060 Exactly. So in this way, we've constructed now a number which must have a new prime factor, 35 00:03:15,060 --> 00:03:20,420 which was known all could be private, something like seven happens to be. 36 00:03:20,420 --> 00:03:26,130 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, 37 00:03:26,130 --> 00:03:30,300 we multiply them all together, we add one and the same argument goes through anyway. 38 00:03:30,300 --> 00:03:35,370 Either our new number is a private self. What's divisible by some new private? 39 00:03:35,370 --> 00:03:40,110 So if I was labouring under the impression I had, there are all the problems in the world in my box. 40 00:03:40,110 --> 00:03:47,460 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. 41 00:03:47,460 --> 00:03:51,870 So I couldn't have had a finite number to begin with. Yeah. 42 00:03:51,870 --> 00:03:58,140 Jamie, perhaps you could tell him we're skipping ahead two millennia now around sort of late 18th century, 43 00:03:58,140 --> 00:04:03,540 there was a conjecture made about how many numbers they were up to a threshold. 44 00:04:03,540 --> 00:04:11,430 OK, so there's this mathematician that's quite well known amongst mathematicians called Carl Friedrich Gauss. 45 00:04:11,430 --> 00:04:17,070 And around the age of 15 or 16, when he was doing various experiments with numbers, 46 00:04:17,070 --> 00:04:21,990 he noticed that if you just did so by experiment, you don't necessarily mean, you know, in a lot of the tests you. 47 00:04:21,990 --> 00:04:26,520 But you mean he's doing all the calculations. He's yes. He's just trying to work out something. 48 00:04:26,520 --> 00:04:29,310 He's doing lots of calculations and looking for patterns. 49 00:04:29,310 --> 00:04:38,730 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, 50 00:04:38,730 --> 00:04:46,050 and you tick the numbers and and plus one and plus two and you go all the way up to a thousand. 51 00:04:46,050 --> 00:04:54,450 And you look at all the primes in that period, that group of numbers as you go as energy changes, 52 00:04:54,450 --> 00:05:04,470 the number of primes in that group of numbers decreases by a factor of about one over the logarithm of any. 53 00:05:04,470 --> 00:05:12,840 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, 54 00:05:12,840 --> 00:05:17,310 because it's going to play a major role in the precise statement of the prime number theorem. 55 00:05:17,310 --> 00:05:20,910 Simon Johnson, explain for us the longer term. 56 00:05:20,910 --> 00:05:27,670 I throw in one number, I get another number out. But what sort of properties does this function the logarithm have? 57 00:05:27,670 --> 00:05:41,200 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. 58 00:05:41,200 --> 00:05:49,390 So. This is roughly speaking, this is the number of digits it takes to write down the number, 59 00:05:49,390 --> 00:05:57,430 for example, the logarithm of ten is one by ten, logarithm of ten is one. 60 00:05:57,430 --> 00:06:04,000 The best ten logarithm of 100 is to the best of a thousand is three. 61 00:06:04,000 --> 00:06:07,540 And what's happening here is that the number of zeros is going up. 62 00:06:07,540 --> 00:06:14,500 You're writing down, you write down the number and the best ten logarithm goes up with within that number of zeros, 63 00:06:14,500 --> 00:06:24,070 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. 64 00:06:24,070 --> 00:06:28,240 You can use you can use a different base instead of ten. 65 00:06:28,240 --> 00:06:35,260 So, for example, you can use some write down in binary, which is how computers store numbers. 66 00:06:35,260 --> 00:06:40,020 And this uses the number two instead of the number 10. 67 00:06:40,020 --> 00:06:46,470 In this case, I got into the base, too, I guess, to take, you know, two to the power of five cents, 68 00:06:46,470 --> 00:06:53,970 thirty to the locker room with kind of the inverse of that processor instead of computing to talk to you two times, 69 00:06:53,970 --> 00:06:57,960 two times to two times, it's about five times being 32. 70 00:06:57,960 --> 00:07:06,860 The longer in about the inverse question is how many times did I have to multiply two by itself to get 30 to. 71 00:07:06,860 --> 00:07:14,570 Yes, exactly, exactly. And the best 10 logarithm of a number is the number of times you have to multiply 72 00:07:14,570 --> 00:07:20,660 10 by itself to get that number so long instead of one hundred two thousand three, 73 00:07:20,660 --> 00:07:28,370 as you said before. Exactly what the prime number theorem uses, a thing called the natural logarithm. 74 00:07:28,370 --> 00:07:42,020 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. 75 00:07:42,020 --> 00:07:49,370 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. 76 00:07:49,370 --> 00:07:55,730 You can't write down numbers using the base E instead of the base ten. 77 00:07:55,730 --> 00:08:01,840 It's, if you like, somewhere in between logarithm to the base two and logarithm to the base three. 78 00:08:01,840 --> 00:08:06,850 It's distinguished by special properties when we come to look at calculus, 79 00:08:06,850 --> 00:08:13,720 when we come to try and differentiate an integrated in some sense just the same kind of object to this, 80 00:08:13,720 --> 00:08:18,460 look to the base 10, the base to in fact, it's related by it's a constant multiple. 81 00:08:18,460 --> 00:08:25,210 So each of these different bases. But it kind of has a special place because of these characters properties, 82 00:08:25,210 --> 00:08:31,600 which we won't describe in too much detail and then show the derivative reacts of the natural 83 00:08:31,600 --> 00:08:38,020 log of X is one over X and you get a very similar relationship with these other log functions. 84 00:08:38,020 --> 00:08:46,510 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. 85 00:08:46,510 --> 00:08:51,070 What was Goussis observation, connecting primes and this logarithm function. 86 00:08:51,070 --> 00:08:59,440 So as I said before, Gauss noticed that when you sort of take these groups of one thousand consecutive numbers and 87 00:08:59,440 --> 00:09:07,690 let's say starting with N as N grows and changes the number of primes in this group of numbers, 88 00:09:07,690 --> 00:09:14,120 I say decreases by approximately one over the logarithm. 89 00:09:14,120 --> 00:09:23,200 Then he kind of conjecture to put forward this idea that perhaps the number of primes less than 90 00:09:23,200 --> 00:09:31,180 a given number might be somehow connected to the logarithm event in a very precise kind of way. 91 00:09:31,180 --> 00:09:35,440 I think if I recall, he kept a diary. 92 00:09:35,440 --> 00:09:41,770 He was very precocious teenager and of insightful comments that people found many years later. 93 00:09:41,770 --> 00:09:53,890 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. 94 00:09:53,890 --> 00:09:59,330 And so if you kind of integrate this up by something over all scales, you get a conjecture. 95 00:09:59,330 --> 00:10:07,180 But the number of primes less than N is going to be about and divided by the natural look of it. 96 00:10:07,180 --> 00:10:14,380 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, 97 00:10:14,380 --> 00:10:19,750 it does tend to infinity with and tells very pretty slowly. 98 00:10:19,750 --> 00:10:24,890 Yeah. Yes. So hang on, let me think for a moment. 99 00:10:24,890 --> 00:10:34,370 If I've done my mental arithmetic correctly, the the natural logarithm is roughly twice the base, 10 logarithm. 100 00:10:34,370 --> 00:10:42,500 Well, very roughly. Very roughly. So the National Laboratory of a million is very roughly 12. 101 00:10:42,500 --> 00:10:48,290 So a million it's a very large number going in, but it's logarithm is very approximately 12. 102 00:10:48,290 --> 00:10:57,380 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, 103 00:10:57,380 --> 00:11:04,400 about a million divided by 12 prime numbers, less than a complete. 104 00:11:04,400 --> 00:11:07,430 The girls can prove this conjecture is an observation. 105 00:11:07,430 --> 00:11:12,520 I think it's quite a romantic story, actually, because he was looking at these tables that he had an opposite pages. 106 00:11:12,520 --> 00:11:17,030 He had a table of the first thousand numbers and a long table. 107 00:11:17,030 --> 00:11:26,270 And it was only because of this proximity of these two piece information that he got drawn into thinking, oh, I wonder how many. 108 00:11:26,270 --> 00:11:31,380 Oh, it looks like these logs on this other side of the page. 109 00:11:31,380 --> 00:11:35,640 We go back to Euclid's argument, it gives us some phone numbers, how many does it give us? 110 00:11:35,640 --> 00:11:41,550 Yes. So as you mentioned, it doesn't give you a verdict like when we started with two or three with no numbers. 111 00:11:41,550 --> 00:11:45,090 And the next one we got was seven. And then after that, we got 43 three. 112 00:11:45,090 --> 00:11:51,630 And obviously we're keeping almost all the numbers like now, lots of prime numbers between what's in the military. 113 00:11:51,630 --> 00:11:56,740 So I think we should get roughly logarithmically many prime numbers in this way. 114 00:11:56,740 --> 00:12:04,350 So that means that, as Jamie was saying, Gousse conjecture that we should have in divided by log in, 115 00:12:04,350 --> 00:12:07,990 which is basic we should think of very nearly every day. 116 00:12:07,990 --> 00:12:12,000 I mean, so it's still a proportion turning to zero because law does tend to infinity. 117 00:12:12,000 --> 00:12:19,350 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, 118 00:12:19,350 --> 00:12:22,650 which is, as Simon was describing, a very small number. 119 00:12:22,650 --> 00:12:28,650 So less than a million that would give you about 12 primes, which is obviously completely wrong. 120 00:12:28,650 --> 00:12:31,310 OK, so we have two problems here. 121 00:12:31,310 --> 00:12:41,070 One is the very precise conjecture of really exactly this many crimes for which even struggling to find more than logarithmically in any crimes. 122 00:12:41,070 --> 00:12:51,060 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. 123 00:12:51,060 --> 00:12:54,330 Jamie, you're going to talk a little bit about intelligence estimates. 124 00:12:54,330 --> 00:13:05,400 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, 125 00:13:05,400 --> 00:13:18,810 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. 126 00:13:18,810 --> 00:13:28,320 Actually, the bonds that he proved were that for some number B, which was which was almost one, 127 00:13:28,320 --> 00:13:42,060 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. 128 00:13:42,060 --> 00:13:48,580 That's about one point two. So he got it like pretty close on both sides. 129 00:13:48,580 --> 00:13:52,320 But I mean, that that's still not quite enough. Two things to say here. 130 00:13:52,320 --> 00:13:56,130 One is that so we might say this is an order of magnitude estimate. 131 00:13:56,130 --> 00:14:06,340 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. 132 00:14:06,340 --> 00:14:10,950 But I think we should now maybe chat about patrician's methods, 133 00:14:10,950 --> 00:14:18,210 but actually just nail down what the precise statement of the prime number theorem is so we can tell how this is, 134 00:14:18,210 --> 00:14:25,350 I mean, further than what we have managed to create. So I'm want to tell us how good an estimate was. 135 00:14:25,350 --> 00:14:27,990 Goussis conjecture. 136 00:14:27,990 --> 00:14:40,680 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. 137 00:14:40,680 --> 00:14:48,510 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. 138 00:14:48,510 --> 00:15:02,910 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. 139 00:15:02,910 --> 00:15:10,190 So the difference, the error, if you like, in this estimate is. 140 00:15:10,190 --> 00:15:15,450 Smaller than any fixed multiple of X on log X. 141 00:15:15,450 --> 00:15:24,250 This error, yours, it can still go to infinity, but just much slower than the maintenance of a log. 142 00:15:24,250 --> 00:15:34,080 Yes, if you like the the ratio of the number of primes up to X to. 143 00:15:34,080 --> 00:15:40,260 The quantity X on Block X will become very close to one when X is large, 144 00:15:40,260 --> 00:15:48,270 so that was the conjecture that was going around to the 19th century, which indeed is now the theory and the prime number theorem. 145 00:15:48,270 --> 00:15:53,490 Going back to Jamie, that's not quite what you have managed to prove. 146 00:15:53,490 --> 00:16:01,980 What trebuchets showed was that this ratio lies between about nought point nine and one point one. 147 00:16:01,980 --> 00:16:11,570 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. 148 00:16:11,570 --> 00:16:20,780 Yeah, actually, it was the first major step towards the actual proof of the what? 149 00:16:20,780 --> 00:16:27,830 There were probably one of the first things I actually showed that it was open to attack. 150 00:16:27,830 --> 00:16:34,220 And now enter Reman Reman versus a single paper on No3. 151 00:16:34,220 --> 00:16:43,820 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. 152 00:16:43,820 --> 00:16:52,340 So one of the most central tools which we introduced is something we call the Raymond Z definition, which is well, 153 00:16:52,340 --> 00:17:01,230 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. 154 00:17:01,230 --> 00:17:05,150 What's the definition of the to function? Right. 155 00:17:05,150 --> 00:17:13,850 The definition of the function evaluated in some point it's called s is that you sum all the 156 00:17:13,850 --> 00:17:21,020 natural numbers from one all the way to infinity and one divided by in to the power of S. 157 00:17:21,020 --> 00:17:23,330 So it s was like two or something. 158 00:17:23,330 --> 00:17:31,400 We could be computing one over one squared plus one of two squared plus one of the three squared plus one of the fourth. 159 00:17:31,400 --> 00:17:37,730 Exactly. And so that's just some number. That's Veeteren two. 160 00:17:37,730 --> 00:17:40,710 Exactly. Is that number. 161 00:17:40,710 --> 00:17:52,260 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, 162 00:17:52,260 --> 00:18:01,990 one over one squared plus one of the two squared and so on, we can factorisation instead as a product over all prime numbers. 163 00:18:01,990 --> 00:18:11,490 Uh, so, so now we have a product of one divided by, uh, one plus one over P squared. 164 00:18:11,490 --> 00:18:15,840 So in a rather similar way that we can write all of them as the product of primes. 165 00:18:15,840 --> 00:18:26,610 In fact, that fact allows us to prove that we can write this infinite sum of a product over primes of these simpler expressions, 166 00:18:26,610 --> 00:18:30,430 one over one minus one over three decades. 167 00:18:30,430 --> 00:18:38,640 Reginald's two. It's one of the P squared oilor talked about this well as one to three. 168 00:18:38,640 --> 00:18:41,730 But Reman extended this for Reman. 169 00:18:41,730 --> 00:18:54,540 What type of object was s so so Reman Reman considered the this Zitter function so called when s was a complex number. 170 00:18:54,540 --> 00:18:59,160 The complex numbers are what are called the real numbers. 171 00:18:59,160 --> 00:19:01,260 All of the decimals, if you like, 172 00:19:01,260 --> 00:19:10,080 PI and 10.1 recurring and all these numbers that you can write out with an infinite number of tickets after the decimal point. 173 00:19:10,080 --> 00:19:17,490 And in addition you throw in the imaginary unit, the square root of minus one. 174 00:19:17,490 --> 00:19:25,230 So this is this is referred to as an imaginary number because there is no decimal number you can write down, 175 00:19:25,230 --> 00:19:32,520 which when you swear you get minus one, no matter what number you write down, if you square it, you get a positive number. 176 00:19:32,520 --> 00:19:41,370 But if you define I to be some unspecified number with one square deal, it's minus one. 177 00:19:41,370 --> 00:19:45,480 You can you can form a perfectly good number system, if you like, 178 00:19:45,480 --> 00:19:55,620 by multiplying I by real numbers and adding it to real numbers to create things like PI plus two. 179 00:19:55,620 --> 00:19:57,510 I for example, 180 00:19:57,510 --> 00:20:07,380 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, 181 00:20:07,380 --> 00:20:14,340 a straight line with zero marks and one marked and every number having a place along this line, 182 00:20:14,340 --> 00:20:23,580 then the complex numbers live in live in a plain, flat, two dimensional space with zero marked and one marked. 183 00:20:23,580 --> 00:20:28,770 And then I marked ninety degrees at a right angle to one. 184 00:20:28,770 --> 00:20:36,540 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. 185 00:20:36,540 --> 00:20:45,600 So you can think in like two axes, you have a horizontal real axis of some U.S., your number line from school, 186 00:20:45,600 --> 00:20:53,230 and then you have another axis, the imaginary axis going a 90 degrees eye on the table in front of me. 187 00:20:53,230 --> 00:20:58,440 I'm crossing my hands at right angles. I you can't see, which is not very helpful to hear. 188 00:20:58,440 --> 00:21:03,530 So you define for us the remains of the functions of infinity as one of what we ask as one or two. 189 00:21:03,530 --> 00:21:09,460 Yes. So. Does that define the function almost entirely? 190 00:21:09,460 --> 00:21:14,710 I'm afraid it doesn't. If that were the case. 191 00:21:14,710 --> 00:21:19,780 So what happened here was that I define it as an infinite sum and not always leave. 192 00:21:19,780 --> 00:21:24,040 Each of the numbers in my song is huge and I keep adding up huge numbers. 193 00:21:24,040 --> 00:21:28,810 I'm just going to get something which just grows and grows and grows and doesn't it doesn't make sense. 194 00:21:28,810 --> 00:21:35,470 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. 195 00:21:35,470 --> 00:21:42,550 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. 196 00:21:42,550 --> 00:21:47,890 So this is like it's horizontal coordinate. Everything of our axes has to be quite far to the right. 197 00:21:47,890 --> 00:21:52,390 It's got to be at least one. The number has to be to the right of the number one. 198 00:21:52,390 --> 00:22:01,060 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. 199 00:22:01,060 --> 00:22:10,840 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, 200 00:22:10,840 --> 00:22:16,330 not to begin with, only if the horizontal part is to the right one. 201 00:22:16,330 --> 00:22:24,370 But then we have a theorem in mathematics which allows us to extend this function also to the left of this one. 202 00:22:24,370 --> 00:22:29,140 So what happens is that my definition, writing it as a work to give you the right answer, 203 00:22:29,140 --> 00:22:34,270 but there's still a unique way of kind of getting a value service to the function, 204 00:22:34,270 --> 00:22:44,170 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, 205 00:22:44,170 --> 00:22:51,700 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. 206 00:22:51,700 --> 00:22:56,100 And it's a fair just mentioned this. It's a continuation, it's saying. 207 00:22:56,100 --> 00:23:03,700 But that is a way I think that is a unique way of building a bit more of a sheet over to the left, 208 00:23:03,700 --> 00:23:10,540 to the left of one, which smoothly continues the spreadsheet. 209 00:23:10,540 --> 00:23:16,060 And so we can actually get a function on on the entire plane. 210 00:23:16,060 --> 00:23:22,690 Apart from one point, there's one point apart from S equals one, which is this point we've been mentioning all the time. 211 00:23:22,690 --> 00:23:28,630 So this is the actual point on the horizontal axis. This is a huge amount of abstract machinery. 212 00:23:28,630 --> 00:23:35,710 OK, so Raymond didn't invent this complex numbers have been going around for about a century before we became law silent. 213 00:23:35,710 --> 00:23:43,900 Could you be, you know, for slightly more explicitly how we can get a handle on the crimes through these different. 214 00:23:43,900 --> 00:23:51,970 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, 215 00:23:51,970 --> 00:23:57,430 but it has an expression as a product of a price and. 216 00:23:57,430 --> 00:24:05,920 There is, in fact, a way to construct from the reproductive function something which has an expression, 217 00:24:05,920 --> 00:24:12,550 as are some overprice or some other powers of prime precision is required. 218 00:24:12,550 --> 00:24:16,910 But to all intents and purposes, one can construct from the retina to function. 219 00:24:16,910 --> 00:24:21,970 Another complex function is another function of the complex plan, 220 00:24:21,970 --> 00:24:29,950 which can be expressed as a sum over, not over all the numbers, but a sum over all the prime numbers. 221 00:24:29,950 --> 00:24:33,590 To be precise, one one takes the logarithm of the Remzi to function. 222 00:24:33,590 --> 00:24:36,910 That one takes the derivative of that function. 223 00:24:36,910 --> 00:24:46,030 So we've constructed this function using calculus by taking a derivative and in fact by using calculus, again, 224 00:24:46,030 --> 00:24:57,760 by taking an integral of this new function, one can pick out the terms involving involving primes between one and X. 225 00:24:57,760 --> 00:25:05,440 And one can go from this inference for all the primes to a sum over primes between one and X. 226 00:25:05,440 --> 00:25:10,330 It's the thing called Pearlman's formula, I guess you're thinking of. 227 00:25:10,330 --> 00:25:17,260 So this is all happening around 1860, but Raymond died quite shortly afterwards in a short, tragic life. 228 00:25:17,260 --> 00:25:21,130 And although he made some great insights into this function, 229 00:25:21,130 --> 00:25:27,100 he did manage to prove that there was a key technical stumbling block to 230 00:25:27,100 --> 00:25:33,080 estimating this integral that would give you the number of times less than Jamie. 231 00:25:33,080 --> 00:25:37,090 Do you want to comment on on what this key stumbling block was? 232 00:25:37,090 --> 00:25:44,710 Well, in order to estimate the integral, which appears when you take the sum of the prime numbers, 233 00:25:44,710 --> 00:25:54,580 less than a quarter X Reman realised that what you needed to do was shift the interval from one line to another. 234 00:25:54,580 --> 00:25:59,170 The process of doing that involves using a result known as coaches' there. 235 00:25:59,170 --> 00:26:03,020 But to do this, the terms which appear inside the interval, 236 00:26:03,020 --> 00:26:11,890 the things that we are integrating into this function involving the logarithmic derivative of the inside function, 237 00:26:11,890 --> 00:26:17,320 needs to satisfy some particular properties inside the region. 238 00:26:17,320 --> 00:26:22,000 Between the two lines that we're moving around interval between that those that property 239 00:26:22,000 --> 00:26:28,900 can be summarised basically by saying that we would like the Remzi to function to be. 240 00:26:28,900 --> 00:26:35,520 Nonzero inside and that region between two lines of integration. 241 00:26:35,520 --> 00:26:43,560 And really, the proof of the prime number theorem was facilitated by high demand and certified to coming 242 00:26:43,560 --> 00:26:54,720 along at the end of the 19th century and showing that this not zero free region existed. 243 00:26:54,720 --> 00:27:02,070 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. 244 00:27:02,070 --> 00:27:08,610 But it took another 35 years until 1896 to show that Zettl one plus it. 245 00:27:08,610 --> 00:27:13,230 So what is the real part? It is the imaginary part that was never zero. 246 00:27:13,230 --> 00:27:20,760 And so, as Jamie described, you can move the integral and the integral is easier to estimate when you move it. 247 00:27:20,760 --> 00:27:29,710 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? 248 00:27:29,710 --> 00:27:37,270 Well, the first proofs that the Raven Zeta function possess this zero for region were relatively long 249 00:27:37,270 --> 00:27:44,080 and difficult compared to the compared to the demonstrations we find in our textbooks these days. 250 00:27:44,080 --> 00:27:51,010 But the underlying idea is much the same. And that's that all one really needs to show, first of all, 251 00:27:51,010 --> 00:28:00,730 is that the and to function has no zeros on a line, on a certain straight line in the complex plane. 252 00:28:00,730 --> 00:28:09,760 And one can one can show this by comparing the values of the Roman to function near three points, 253 00:28:09,760 --> 00:28:16,690 one where it has this is a zero that we would like to prove doesn't exist. 254 00:28:16,690 --> 00:28:25,090 One is at the point one near the number one where we know that the Roman Zeta function is very large. 255 00:28:25,090 --> 00:28:34,570 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. 256 00:28:34,570 --> 00:28:43,390 And that's the third point. If the Regency to function has a zero sum with real one. 257 00:28:43,390 --> 00:28:48,190 So if there is indeed a bad point with the logarithmic derivative of the Ravens, 258 00:28:48,190 --> 00:28:53,410 they did function, has a has a poll, and we need to move this integral part past the poll. 259 00:28:53,410 --> 00:29:06,480 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, 260 00:29:06,480 --> 00:29:13,740 a poll with real part one and it Macary part twice the imatinib part of the zero. 261 00:29:13,740 --> 00:29:19,860 And we know that this isn't so. We know that the only poll of the frequency to function is that one. 262 00:29:19,860 --> 00:29:25,650 And this will this will enable us to to prove by contradiction that it's not possible for the presidency to 263 00:29:25,650 --> 00:29:32,950 function to have a zero real parts equal to what's in the details of the argument that Simon has just mentioned. 264 00:29:32,950 --> 00:29:43,110 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. 265 00:29:43,110 --> 00:29:48,870 And I remember thinking when I was there during my Masters that MacGraw almost view 266 00:29:48,870 --> 00:29:54,630 almost the prime number theorem as a corollary of the cosine double angle formula. 267 00:29:54,630 --> 00:29:58,680 OK, so we reached nineteen hundred and we proved the problem with him. 268 00:29:58,680 --> 00:30:07,330 And the curious state of affairs was that people began to realise that this complex analysis approach that Simon Cy-Fair and Jamie 269 00:30:07,330 --> 00:30:16,620 have been describing was not just one particular method that they had managed to successfully use to prove the point of the theorem, 270 00:30:16,620 --> 00:30:22,890 but in some ways was equivalent to the prime no theorem in a very precise sense that the prime number theorem was 271 00:30:22,890 --> 00:30:32,010 equivalent to showing that there were no zeros or the Remzi to function on the one line with real parts equal in one. 272 00:30:32,010 --> 00:30:37,350 And so the general consensus was that this complex analysis machinery was completely 273 00:30:37,350 --> 00:30:42,120 necessary to prove the prime number theorem because it was equivalent to it. 274 00:30:42,120 --> 00:30:47,950 So there could never be what physicians call an elementary proof. 275 00:30:47,950 --> 00:30:54,180 So there was this assumption, correct. So it turned out, rather surprisingly, that this assumption was not correct. 276 00:30:54,180 --> 00:31:04,410 In 1948, 49, a Norwegian mathematician named Unassembled, came along and in fact did give an elementary proof of the prime number theorem. 277 00:31:04,410 --> 00:31:08,380 And by elementary, you mean Sarabhai Elementary. 278 00:31:08,380 --> 00:31:15,540 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. 279 00:31:15,540 --> 00:31:18,030 It doesn't get mentioned a single time is proof, 280 00:31:18,030 --> 00:31:25,080 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. 281 00:31:25,080 --> 00:31:29,640 And how does the proof go rough? It splits into a couple of stages, right. 282 00:31:29,640 --> 00:31:37,710 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, 283 00:31:37,710 --> 00:31:47,220 which instead of considering sums of primes so Asama over primes, we are considering products of two primes up to two primes. 284 00:31:47,220 --> 00:31:51,180 And it turns out rather crucially, that in this approach, 285 00:31:51,180 --> 00:31:59,850 the term which should come from the zeroes of the Remzi to function in the kind of original proof completely cancels. 286 00:31:59,850 --> 00:32:04,800 So we don't actually need to know anything about the zeros the other and the council and that's it. 287 00:32:04,800 --> 00:32:06,270 And they have a second part of the proof, 288 00:32:06,270 --> 00:32:14,820 which is to actually show that these statements about sums of products of primes is strong enough to give the original prime number theorem. 289 00:32:14,820 --> 00:32:21,630 And that is a elementary but rather involved argument, if I remember correctly, right, Jamie? 290 00:32:21,630 --> 00:32:29,280 There was quite a controversy at the time and so many years later between Salberg and a Hungarian mathematician called Paul Aatish, 291 00:32:29,280 --> 00:32:35,880 who also claimed to have found this elementary proof around the same time, Paul, politicians, 292 00:32:35,880 --> 00:32:40,650 a lot of mathematicians will know who is quite a prolific publisher of material. 293 00:32:40,650 --> 00:32:46,200 He is well known enough that in a similar way to which we have back in numbers for actors, 294 00:32:46,200 --> 00:32:55,080 we also have Aatish numbers from mathematicians and he claimed that through collaboration with other outlets. 295 00:32:55,080 --> 00:33:05,280 So sorry. And so it's if there is a Norwegian pronunciation expert, I'll just refer to themselves and ask them that. 296 00:33:05,280 --> 00:33:15,150 In collaboration with Sjoberg, he could also claim credit for the proof that Sjoberg provided of the prime number theorem using elementary 297 00:33:15,150 --> 00:33:26,130 methods because he claimed that without some communication or idea from him that he had given to Salberg, 298 00:33:26,130 --> 00:33:30,810 Salberg would not have been able to complete the second part of the proof. 299 00:33:30,810 --> 00:33:44,220 And there's an apocryphal story surrounding this sort of feud whereby Salberg is supposed to have overheard in some university maths department. 300 00:33:44,220 --> 00:33:54,300 Another mathematician saying that Paul Aatish and some Norwegian fellow had found an elementary proof of the prime number theorem 301 00:33:54,300 --> 00:34:04,420 to which he strongly objected and and went ahead and published his proof of the prime number theorem under his own name. 302 00:34:04,420 --> 00:34:10,900 There's an article online by by Goldfields that tries to clear up all the correspondence 303 00:34:10,900 --> 00:34:16,120 about who came first and what information passed between who I trust and SALBERG, 304 00:34:16,120 --> 00:34:20,950 which I invite you to read, but I'm not sure it completely clarifies the issue. 305 00:34:20,950 --> 00:34:22,820 To close, as I mentioned in my introduction, 306 00:34:22,820 --> 00:34:29,640 that the problem with there was at the heart of one of the big unsolved problems in mathematics today, Simon Johnson. 307 00:34:29,640 --> 00:34:31,760 Tell us what that problem is. 308 00:34:31,760 --> 00:34:41,780 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, 309 00:34:41,780 --> 00:34:50,420 one needs to show that the the agreement Zitter function does not vanish at any point on a certain line in the complex plane. 310 00:34:50,420 --> 00:34:55,760 And the the Riemann hypothesis is a contract, 311 00:34:55,760 --> 00:35:04,850 a description of where all the zeroes of the agreement theta function are all the points, whether in such a function vanishes. 312 00:35:04,850 --> 00:35:11,090 And the conjecture is that apart from some some very predictable, boring, 313 00:35:11,090 --> 00:35:17,930 well understood zeros, all the zeros lie on the on the line real party equals one half. 314 00:35:17,930 --> 00:35:24,050 So this is a vertical line in the complex plane, a distance one half to the left of this. 315 00:35:24,050 --> 00:35:28,220 This real path equals one where we prove there are no zeros and that the number theorem, 316 00:35:28,220 --> 00:35:35,000 the conjecture is that in fact, all the zeros lie a distance one half to the left of that line. 317 00:35:35,000 --> 00:35:41,240 If we want to translate this property of the Regency to function back into some property of prime numbers, 318 00:35:41,240 --> 00:35:50,030 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, 319 00:35:50,030 --> 00:35:57,350 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, 320 00:35:57,350 --> 00:36:02,480 but about that size at the moment, 321 00:36:02,480 --> 00:36:09,800 you would become a very famous mathematician indeed if you could show that the error was almost most X to the power. 322 00:36:09,800 --> 00:36:19,600 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? 323 00:36:19,600 --> 00:36:28,000 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 324 00:36:28,000 --> 00:36:30,460 of the prime number there and there is a proof of the prime number theorem, 325 00:36:30,460 --> 00:36:38,890 which although it is perhaps complicated, certainly a little difficult to understand at first I personally struggled to understand it. 326 00:36:38,890 --> 00:36:50,490 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. 327 00:36:50,490 --> 00:37:02,010 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, 328 00:37:02,010 --> 00:37:08,160 but it might in some ways still be the best way to understand it. 329 00:37:08,160 --> 00:37:15,090 These connexions between different areas of maths in this case, between complex analysis and number theory, 330 00:37:15,090 --> 00:37:27,840 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, 331 00:37:27,840 --> 00:37:35,670 that are complex analytic formulation of the problem number theorem. 332 00:37:35,670 --> 00:37:43,980 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, 333 00:37:43,980 --> 00:37:52,980 I think one could argue fairly convincingly that it's connected, for example, to Andrew Wiles, proof of last there. 334 00:37:52,980 --> 00:37:58,620 Well, thank you very much, Sophia. Jeremy Simon, we've covered a lot of ground in 45 minutes. 335 00:37:58,620 --> 00:38:03,381 The next week in our spare time, we'll be talking about Shakespeare and music.