We embark on a journey into the world of L-functions, by introducing the Riemann Hypothesis and the dream of a new geometry over the “field with one element”.The aim of RH Saga Season 1 is to map the landscape of L-functions, as a foundation for future in-depth exploration of some of the most immortal math problems of all time.

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Introduction to the RH Saga

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The Riemann hypothesis,

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may be the greatest unsolved math problem of all time,

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but it is only a small part of a much bigger story,

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and this story is the quest,

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for a new geometry underlying the theory of L-functions.

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♪ (intro music) ♪

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In the RH Saga,

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we will explain what L-functions are,

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explain the dream of this new geometry,

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that some people call geometry over the field with one element, or F1,

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we’ll explain the connection, to the Riemann hypothesis,

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and other immortal problems,

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like, the BSD conjecture and the Langlands program.

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This will be a long journey.

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The starting point is the theory of L-functions.

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Where the simplest example,

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is, the Riemann zeta function.

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The dream is to find this unknown, hidden, elusive, F1 geometry.

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The hope, is then, that this geometry,

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could open up a path to a proof,

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of the Riemann hypothesis.

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Today, I want to show you an article of Manin,

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as an introduction to F1,

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then give a quick recap of the Riemann hypothesis,

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and finally say something about how you, or I,

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could start the search for a proof.

Introduction to Episode 1: The Dream

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♪ (music) ♪

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But, before we dive into that,

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let me just say that if you feel, in season one,

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that you LOVE this kind of math,

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it’s so beautiful, but there are too many words you don’t understand,

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There are too many technical things,

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and you just feel a bit lost,

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then do not freak out,

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this is how it’s supposed to be.

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I promise that starting next season, season two,

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I will do my very best to define everything carefully,

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and explain things properly.

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The point of season one,

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is only to build this global overview

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which will help us, together, later,

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to connect the many different aspects

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of this enormously large concept space

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that we’re trying to make sense of.

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Everything we do in the RH Saga,

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is built on conversations between myself,

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and a very good friend whom you will meet in due time,

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and for the two of us,

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the dream is really to find this new geometry,

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and maybe even prove the Riemann hypothesis.

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But what really matters is not the end goal,

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but the journey.

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I hope you’ll join us,

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for this voyage into the most mysterious,

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and beautiful parts of the mathematical landscape.

Chapter 1: Intro to F1

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For a first encounter with F1,

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I just want to show you this article of Yuri Manin.

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Manin is a legend,

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sadly he passed away recently.

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He was one of very few mathematicians in the world,

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who possessed something like an overview of all of mathematics.

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In this article,

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I just want to point out a few key ideas,

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that we will encounter later.

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And convey to you,

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that when I say “everything is connected”,

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I am not making things up.

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And you can read this article yourself later,

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it’s openly available on the arXiv.

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The title here, is “Numbers as functions”,

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and the idea,

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is that in order to solve the deepest problems of number theory,

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we may have to reimagine,

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the very core of mathematics itself.

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In particular, we may have to rethink,

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what we mean by the word “number”.

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Manin discusses a specific way of doing this,

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following a theory developed by Alexandru Buium.

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Here in the abstract,

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you see this phrase,

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“geometry over fields with one element”.

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He also mentions the very unexpected idea,

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that there may be deep connections,

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between prime numbers, and physics.

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The “p” here, is a prime number.

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The entire article is a story,

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making unexpected connections

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between many different ideas.

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Let’s just skim through the article together,

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and I will highlight some of the ideas that later,

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will become clues for us,

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in the sort of “detective story”,

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of pursuing F1-geometry.

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To begin with,

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here is perhaps the most famous of all math formulas,

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e to the pi i equals minus one.

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He also mentions,

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the special value of the Riemann zeta function,

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pi-square over six,

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which you may have seen,

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in the connection with the Basel problem.

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The discussion here in the beginning,

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focuses on a certain class of numbers,

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which pop up in quantum field theory,

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for mysterious reasons.

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These numbers are called periods.

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On page three,

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we encounter this super important idea,

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which is roots of unity.

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And then early on page four,

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he mentions this hope that we have,

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of approaching the Riemann hypothesis.

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Some interesting examples of periods are algebraic numbers,

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so zeros of polynomial equations,

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the number pi is a period,

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and then these strange numbers,

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you get by applying the gamma function to a rational number.

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So we could call them fractional gamma values.

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So everything here is super interesting,

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the last few things I want to mention are these Feynman integrals,

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or Feynman path integrals,

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connected to amplitudes in quantum field theory.

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And, on page 14, these notions called

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Grothendieck rings and Witt rings.

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And they are examples of algebraic structures,

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with something called “lambda-operations”.

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On page 16, we see something called Fq,

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which is a finite field, we’ll come back to that

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and this symbol F1, which is at the heart of this mysterious geometry.

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Finally on page 18, he uses this phrase,

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“the unfathomable abyss”,

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and I really like this phrase,

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because it conveys something of the depth you fell that you encounter,

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in this space of ideas.

Summary of Chapter 1

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♪ (gentle piano) ♪

Chapter 2: Recap of RH

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Okay, let’s do a quick recap of the Riemann hypothesis.

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The prime numbers are the numbers

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2, 3, 5, 7, 11, 13, 17, 19, and so on.

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If you write down the divisors,

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of each positive integer,

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You’ll see the prime numbers,

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as the numbers with exactly two divisors.

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Let’s look at some larger primes.

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For example, here are the primes just above one hundred.

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It’s 101, 103, 107 and 109.

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Then it’s 113, and then, it’s 127.

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Followed by, 131, 137, 139, and 149.

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If you look at the jumps from one prime to the next,

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they seem to be rather random.

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Sometimes two primes are immediate neighbors.

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“Twin primes”

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Like 101 and 103.

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But then sometimes, there are huge gaps,

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like between 113 and 127.

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Is there an underlying structure here?

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Can we in general predict or understand when the next prime appears?

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This is where the Riemann zeta function enters the story.

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The prime numbers are connected to the zeros of the Riemann zeta function.

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The zeta function, is a function from complex numbers to complex numbers.

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This means,

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that for any complex number you give as input,

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you get a complex number as output.

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For example, input two, gives output pi-square over six.

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Input minus one, gives output, minus one over twelve.

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And, input minus two, gives output zero.

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So the number minus two,

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is a zero of the Riemann zeta function.

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If you look at all these zeros as points in the complex plane,

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some are quite easy to compute.

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They are -2, -4, -6, and so on.

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The negative even numbers.

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These are called the trivial zeros.

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The other zeros, called non-trivial,

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are all contained in the so-called critical strip.

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Meaning, that the real part,

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is between zero and one.

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The Riemann hypothesis states,

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that the non-trivial zeros,

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all have real part exactly equal to one half.

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In other words, that they lie on the so-called critical line.

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Mathematicians have computed,

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the first twelve trillion of these non-trivial zeros,

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and checked that they, at least, lie exactly on the critical line.

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So, their real part is one half.

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What about the imaginary parts?

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Well, you can compute them,

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and for the first few zeros,

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the imaginary parts are,

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around 14.1, around 21, around 25,

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this sequence of numbers is called the Riemann spectrum.

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And, we can compute a longer list of these values using Sage.

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You could install Sage on your own computer,

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but if you want to get started real quick,

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go to sagecell.sagemath.org

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and type the following code.

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This code just prints the first ten elements of the Riemann spectrum.

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But, using Sage, you can get any number,

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up to around two million.

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To get a first idea why these zeros are so interesting,

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let’s look at the function,

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f of x equals minus cosine of 14.1 times log x.

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This is a cosine wave, with 14.1 as the angular frequency,

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and log x as the variable.

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And “log” here, is the natural logarithm,

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which you may have seen as “ln”.

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Let’s plot the graph of f,

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from x equals 1 to 15.

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This looks like some wave,

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where the wavelength increases with x,

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but, look at the peaks!

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It’s not perfect, but there are peaks close to one,

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close to 2, 3, 5, and slightly above 7.

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And there is one last peak here between 11 and 12.

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Let’s now add one more term to the definition of f.

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So now f of x is minus cosine, of 14.1 log x,

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minus cosine of 21 log x.

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This function has peaks, roughly at,

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1, 2, 3, 5, 7, and then slightly above 12.

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Now we keep going, with more and more terms.

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Using the numbers from the Riemann spectrum,

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to build these cosine waves.

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And you can do this yourself, in SageMath.

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with ten terms,

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there are very clear peaks,

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at 1, 2, 3, 5, 7, 11, and 13.

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But, we also see the beginning of some smaller peaks.

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We can do any number of terms here,

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let’s use one hundred terms.

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Let’s extend this graph all the way up to x equals twenty.

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Now, maybe you can guess what the smaller peaks are.

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Following two, there are peaks at 4, 8, and 16.

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Following three, there is a peak at 9,

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and there would be more peaks at 27, 81, and so on.

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These are the powers of two, and the powers of three.

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In general, as we add more and more terms,

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and as we extend the range of x-values,

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there will be spikes at all prime powers.

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And a prime power,

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is just a prime number raised to some positive exponent.

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These graphs are a first tiny hint,

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that although we usually focus on the primes,

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the prime powers will also be super important to us.

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And if you want to look at these graphs,

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maybe with a different number of terms or different range of x-values,

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here is the code that we used on the sage-cell website,

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to generate these plots.

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Just change the number 10 and the number 15 if you want.

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Just recall, that these graphs,

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all these spikes,

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came not from knowing the prime numbers,

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they came only from the Riemann spectrum.

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So clearly, there is some kind of connection here,

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between the Riemann spectrum,

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and the prime numbers.

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And this is just insane!

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The prime numbers shouldn’t have anything to do with cosine waves.

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If you dive more into the details of this connection,

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which we’re not doing today,

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and you want to understand how many primes there are

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up to a given number,

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like up to twenty or up to one million,

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then the key thing turns out to be,

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how far from the critical line a zero can occur.

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And the most perfect and beautiful situation would be,

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if all the zeros were actually exactly on the line.

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And this conjecture,

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is the Riemann hypothesis.

Chapter 3: Proof of RH?

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Now, how could we prove the Riemann hypothesis.

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This is, of course, the million dollar question.

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There are many books about the Riemann zeta function,

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and lots of books about the Riemann hypothesis.

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In some of these,

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you will find lists of lots of things that people have tried,

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and lots of ideas, that might possibly play a role in some future proof.

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Of course, no one really has any idea,

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as to what will, eventually, work.

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But, I think, one of the best places,

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maybe the best place,

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if you want to read about these ideas,

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is a quite recent article of Brian Conrey.

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Conrey is one world’s leading experts,

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on the Riemann hypothesis.

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He has lots and lots of super interesting research papers,

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but for the average person on the street,

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probably the most famous paper is the one

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where he proved that at least 40 percent of the zeros

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in the critical strip, are actually on the critical line.

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You can read this paper yourself,

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or skim-read it,

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I just want to highlight a few of the ideas that I think are most interesting.

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There is a discussion in the beginning,

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of the history behind the problem,

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and the connection between the primes and the zeros.

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There is a short section on

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“Why do we think RH is true?”,

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that is also super interesting.

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Then you have this spectral interpretation idea,

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meaning that the zeros may be eigenvalues of some operator.

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Some initial thoughts about proving RH,

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that sounds interesting,

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then I want to highlight this number 16,

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Weil’s explicit formula and positivity criterion.

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17 is Li’s criterion,

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and then you have the analogy with function field zeta functions.

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Also, look at the notion of the Selberg class.

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That’s an attempt to write down axioms for what an L-function is.

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This idea of a family of L-functions is important,

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and again, the word “positivity” appears.

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Positivity is really a key stepping stone,

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I think, to any future proof of RH.

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Finally, the last section, or second to last, random matrix theory,

21:35

that is a super interesting field of research.

21:39

Okay, check out the paper,

Summary of Chapter 3

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to summarize,

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maybe the three most important ideas here,

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number one, we should study all L-functions,

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not just the Riemann zeta function,

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and that is because there are patterns and phenomena,

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which are not visible if you study the Riemann zeta function only.

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Number two, focus on this notion of positivity.

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This is one of few general approaches,

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to the Riemann hypothesis,

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which works for L-functions in general,

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and not just the Riemann zeta.

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Number three, revisit the analogy with function fields.

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The analogy with function fields zeta functions,

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where the Riemann hypothesis is already proven,

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is one of the main reasons we believe the Riemann hypothesis is true.

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It’s also one of the main sources for new ideas,

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for a hidden F1-geometry.

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In the next episode,

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We’ll start looking at sources of L-functions.

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L-functions come from something,

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we call primal objects.

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And we will also begin,

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looking at another one of the great problems,

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namely the Langlands program.

23:16

♪ (outro music) ♪