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  • yes so we're going to talk about L-functions. I want to give you an example

  • first of all of one L-function that I think you probably would recognize and

  • this is the Riemann zeta function, which depends on a complex number

  • s is x+iy. 'i' here is the square root of -1, and you can define it either by

  • a sum over the integers, all the integers, all the positive ones. Or, this is a great discovery of Euler:

  • I can write it as a product over the prime numbers. and so the zeta function

  • encodes information about the primes. It contains all the primes and if you can

  • unravel how the information in the zeta function is encoded you can determine

  • properties of the primes. That's one of the L-functions, and let me tell you

  • a little bit more about the Riemann zeta function because that's gonna be part of the

  • story. So the Riemann zeta function is defined as a sum over the integers or a

  • product over the primes, and these formuli work if x is greater than one.

  • But it turns out you can find formuli that match these when x is greater than 1.

  • When x is less than one work as well. And here's what the Riemann zeta function

  • looks like. So these formuli work to the right of the line x=1. So they work

  • out here. And they tell you the value of the zeta function anywhere in this in this

  • region. Now it turns out the zeta function has this amazing symmetry

  • which was discovered by Riemann, which say that

  • the line that I have drawn in blue there that passes though the point one half, that's

  • a symmetry line of the Riemann zeta function. If I know the value the zeta

  • function at some point here, I reflect that point through the blue line to get

  • a point here and know its value here. I can deduce that very easily. So Riemann reflection

  • formula, the symmetry formula, tells us that because I know he zeta

  • function in this range I also know in this range as the reflection so that

  • just leaves this strip which we don't understand. This trip is where the

  • function is really mysterious. We know that in this strip,

  • there are infinitely many points where the zeta function takes the value 0.

  • The belief is - and this is the Riemann hypothesis - that these points where the

  • Zetas function

  • takes the value zero, all lie exactly on the symmetry line, and this is called the

  • Riemann hypothesis. And this is one of the great mysteries of mathematics.

  • Riemann hypothesized this in 1859 and it's been tested hugely since.

  • BRADY: professor, it's well-known in mathematics that if any mathematician can prove the

  • Riemann hypothesis, he or she is destined for greatness.

  • PROF: Yeah. BRADY: Why?

  • PROF: Well, because the Riemann zeta function encodes information about the

  • prime, so if we know the positions of the zeros, that tells us really important

  • information about the prime numbers.

  • BRADY: What information? What does it tell you that you don't already know?

  • PROF: it tells us things we already suspect, because we sort of

  • believe the Riemann hypothesis is true, but there are things we can't prove. So for example,

  • how many primes are there up to a trillion? We have a formula that we believe is true but we

  • can only prove that formula if we know the Riemann hypothesis is true.

  • BRADY: So it's like a whole bunch of houses would suddenly have a foundation. PROF: Absolutely,

  • and there are whole books written assuming the truth of the Riemann hypothesis,

  • so the foundations of our understanding of the primes would disappear if the Riemann

  • hypothesis turned out to be false. It's known there are infinitely many zeros of the

  • Riemann zeta function inside the critical strip, and it's known that at

  • least 40% of those do lie on the symmetry line. It is known, and this is a

  • result of hugely extensive computations that the first 10 trillion zeros

  • lie exactly on the symmetry line. That's the first ten thousand billion. It's known that

  • batches of zeros, much higher up, beyond the 10 trillion, lie on the line,

  • so I think the world record at the moment is there's a batch of zeros up beyond the 10^36th zero,

  • That is the billion billion billion billion-th zero, somewhere up there

  • there's a whole batch which we know lie exactly on the line, so people have put a

  • pretty huge effort into this, and there's a long history of this:

  • Alan Turing, when he built the first electronic computer, one of the first things he did

  • was compute the zeros of the zeta function. He found about the first 1000 lie on the line.

  • BRADY: professor, if a mathematician one day finds a zero in the strip but not on the line,

  • is that mathematician gonna be a hero or a pariah?

  • PROF: [laughs] Well they would certainly be very famous, and I think mathematicians

  • very sanguine about this, they'll just accept the truth, they want to know the

  • truth, that person would be famous, but I think it would be a kind of

  • ugly kind of fame, not a beautiful kind of fame.

  • So we got these properties and this is what I want to emphasize: we can write the zeta

  • function as a sum over the integers, or as a product over the primes,

  • and the zeta function has a symmetry line, reflection symmetry line,

  • and we believe that all the zeros of the zeta function around that line

  • lie exactly on that. This is one of the great mysteries in mathematics, and people have

  • spent 150-odd years thinking about this. And one natural way

  • if you're given a mystery is to try to understand: well, is this a more general

  • property? So are there other functions which look like the Riemann zeta function?

  • And maybe by finding lots of those functions and comparing them that tells you the

  • essential properties that make the Riemann hypothesis true. Maybe it allows

  • us to prove it. So an example would be: if you think about evolution,

  • Darwin wanted to establish evidence for evolution, so he goes to the Galapagos Islands,

  • and he has to find lots of finches that're all very similar but not

  • identical, and by making comparisons of the similarities and the differences you

  • then understand the essential properties that led them to evolve as they did.

  • So we want to find the cousins of the Riemann zeta function. To see whether

  • there are any, what might you do? Well you might look at this sum over the integers,

  • and you might change some of these plus signs into minus signs and see, well,

  • can you have any combination of pluses and minuses here? And it turns out

  • it's very rare that by changing some of these plusses into minusses you can write

  • this sum, the resulting sum, as a product of the primes

  • but be slightly different to this product but would resemble it. and you'd have something

  • that had a symmetry line that's very very rare that that happens. to there is

  • one example of one. let me go back to the Riemann zeta function going to split the

  • primes up into two classes: 2 will always be special believe that is it is

  • but the odd primes

  • will either be - if you divide them by 4 - either remainer of 1 or 3.

  • So you divide 3 by 4,

  • the remainders 3. Divide 5 by 4

  • The remainder's 1. Because 5 is 4 + 1.

  • 7: divide that by 4, uh the remainder is 3.

  • 11 is 3. and 13 is 1.

  • So the ones that are divisible

  • where the remainder's 3 will flip their sign, so we put that one there

  • We'll put a 1 there,

  • a 1 there, and we'll do the other 2.

  • And that gives you an L-function which has a symmetry

  • property to expand these products out you get a sum of the integers exactly like the

  • Reiman Zeta function so this is an example of a function which is an L-function.

  • a lot of these were discovered in the nineteenth century. There are infinitely many

  • of them

  • BRADY: You said they were rare

  • Ya ya ya, um. despite the fact that there are many

  • they're extremely rare amongst all possible combinations of

  • plusses and minusses that you can have there.

  • So we know there are infinitely many other

  • functions that look like the Reimann Zeta function. you can write them as a sum

  • of the integers, you can write them as a product of the primes - they have a

  • symmetry line - and we believe they all have a Riemann hypothesis. So that was the

  • 19th century then in the 20th century people found other examples of

  • functions that have those properties you can write them as sums over the integers,

  • or products over the primes, they have a symmetry line, and we believe that they

  • a Reimann Hypothesis. And one of the heroes in this story's Ramanujan.

  • Sir Ramanujam was studying the following function:

  • X times (1- X) to the power of 24.

  • (1-X squared) to the power of 24.

  • times (1-X cubed) to power 24 times (1 - X to the 4th)^24 etc

  • you get the point and what he discovered was that he could write this as a sum of

  • powers of X. (1 times X) minus (24 times X squared) you can do at home

  • multiply this thing out. that's 24 times X squared plus 252 times X cubed minus

  • 1472 times X power 4 etc...

  • and this keeps going on and you get these beautiful

  • whole numbers coming out now here's the miracle.

  • 2 times 3 = 6

  • -24 times 252 = -6048. So these numbers have the

  • property that for example three times five is 15 and if I go out to the 15th

  • number in this series

  • the the coefficient there,

  • the integer that appears is 252 times 4830.

  • This meant you could associate an L-function with these

  • things. so here's what you do you write 1 - 24 over 2^s, + 252 over 3^s,

  • minus 1472 over 4^s, + 4830 over 5^s, et cetera and this series follows

  • from Ramanujan's observation a couple more observations that I won't sort of explain

  • in more detail it follows that you can write this series as a product over

  • primes just as we did for the Riemann zeta function you can plot that in the

  • complex plane and he has a symmetry line exactly like the Riemann zeta function

  • does and the zeros of this function are believed to satisfy Riemann hypothesis.

  • Now these functions are a class of functions called modular forms. They have certain

  • symmetry properties which we believe give rise to the kinda general patterns

  • that you see in the Riemann zeta function there is a symmetry line and

  • zeros on that line a Riemann hypothesis. and these have been studied throughout the

  • twentieth century and Andrew Wiles in his great work on Fermat's Last Theorem

  • used this connection between modular forms that its functions like this and

  • L- functions that its functions like this to proove Fermat's Last Theorem.

  • For all these L-functions we believe there's a Riemann hypothesis so the big goal is to

  • say well can we find a pattern

  • can we see any similarities that would give us a clue as to why the Riemann

  • Hypothesis is true, for any of them? and one of the things that's been done

  • recently is to produce a huge database of millions of these L-functions with

  • all the properties tabulated in a very clear and simple way and basically the

  • idea is to throw this out to the world and say can you help us find the pattern?

  • and what you'll find there is the Riemann zeta function, you'll find

  • Ramanujan's L- function, you'll find the L functions that Andrew Wiles studied,

  • you'll find even more exotic and weird L-functions that we believe all have a

  • Riemann Hypothesis. and you go to this website you find these zeros, you'll find plots of

  • these L-functions ... What's the pattern?

  • BRADY: if I go to this database though, I'm not gonna make a

  • breakthrough, am I? Surly this is only a resource for mathematicians

  • Well...

  • It's originally for mathematicians, but I think these properties are accessible to a

  • wide range of people this database is set up so if you click on any technical

  • word, up comes an explanation of that word now you put some mathematics to

  • understand that but looking at the pictures I something anyone can do and

  • somewhere in there is a pattern that mathematicians haven't spotted. so it may

  • well be that somebody else spots it. not likely but it is possible

  • too big for this the right knees heavyset said it and I'll say it again

  • this is a prime between and into an agreement did precisely that he

  • explained how to extend the this function to all possible values except

  • for one so there's only one billion there is nothing you can do

yes so we're going to talk about L-functions. I want to give you an example

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黎曼假說的關鍵 - Numberphile(數字愛好者) (The Key to the Riemann Hypothesis - Numberphile)

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    林宜悉 發佈於 2021 年 01 月 14 日
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