Introduction

Introduction

I have been told that this Lectorium of the Polytechnic Museum has been a venue for many great names in modern Russian culture, including poets: Mayakovskiy, Blok, Yevtushenko, Voznesensky, and others.

It is a great honor to speak at a place associated with such immortals; but I have the superstitious feeling that I should try to win a little favor with their ghosts before proceeding with my talk.

In a spirit of humility, therefore, I offer a few apt lines from Boris Pasternak. I beg you, and any ghosts who may be listening, to excuse my poor Russian.

Есть в опыте больших поэтовЧерты естественности той,Что невозможно, их изведав,Не кончить полной немотой.

В родстве со всем, что есть, уверясьИ знаясь с будущим в быту,Нельзя не впасть к концу, как в ересь,В неслыханную простоту.

Но мы пощажены не будем,Когда ее не утаим.Она всего нужнее людям,Но сложное понятней им.

The Riemann Hypothesis

A Great Unsolved Mathematical Problem

presented by

John Derbyshire

author of Prime Obsession (2003) and Unknown Quantity (2006)

Mr. Derbyshire’s Math Books

Mr. Derbyshire’s Math Books (cont.)

Prime Obsession (2003)

• All about the Riemann Hypothesis

• Mixes math with history and biography

• Awarded the 2007 Euler Prize (“for popular writing on a mathematical topic”) by the Mathematical Association of America.

Unknown Quantity (2006)

• A history of algebra for non-mathematicians

• Published in paperback May 2007

• Includes references to Riemann’s work in function theory and topology

Mr. Derbyshire’s Most Recent Book

We Are Doomed: Reclaiming Conservative Pessimism (2009)


All nontrivial zeros of the zeta

function have real part one-half.

Q: What is the RH about ?

A: It is about prime numbers.

Prime Numbers

• A prime number doesn’t divide by anything (except itself and 1).

• 63 divides by 3, 7, 9, and 21, so 63 is not a prime number.

• 29 divides by … nothing (except 29 and 1), so 29 is a prime number.

• First few prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 2929, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101,

…

More Prime Numbers

There is an Infinity of PrimesDo the primes go on for ever?

Yes: the Greeks proved it.

• Proof :Suppose there were a biggest prime, P.Form this number: (12345 . . . P) + 1.Plainly it is bigger than P.Yet it is not divisible by P, nor by any smaller number. If you try, you

always get remainder 1!Therefore either it is not divisible by anything at all, or the smallest

number that divides it is bigger than P.In the first case it’s a prime bigger than P; in the second, its smallest

factor is a prime bigger than P.Since these both contradict my original “suppose,” my original

“suppose” must be wrong. There is no biggest prime.

Unsolved Problems About Prime Numbers

• Goldbach’s ConjectureEvery even number after 2 is the sum of two prime numbers (e.g. 98 = 19 + 79).

• The Prime Pair ConjectureThere are infinitely many pairs of primes just two apart (like 41 and 43).

• The Riemann Hypothesis

Topic 1

*

The Sieve [Решето] of Eratosthenes

Eratosthenes of Cyrene

• Greek, 276-194 B.C.• Cyrene is in today’s

Libya, then part of post-Alexander Greek Egypt under Ptolemy II.

• All-round intellectual: good achievements in philosophy, astronomy, geography, drama, ethics.

• Accurately measured the Earth’s circumference, and the tilt of Earth’s axis.

The Sieve of Eratosthenes

2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

100 101 102 103 104 105 106 107 108 109

Sieving out the 2’s22 3 44 5 66 7 88 9

1010 11 1212 13 1414 15 1616 17 1818 19

2020 21 2222 23 2424 25 2626 27 2828 29

3030 31 3232 33 3434 35 3636 37 3838 39

4040 41 4242 43 4444 45 4646 47 4848 49

5050 51 5252 53 5454 55 5656 57 5858 59

6060 61 6262 63 6464 65 6666 67 6868 69

7070 71 7272 73 7474 75 7676 77 7878 79

8080 81 8282 83 8484 85 8686 87 8888 89

9090 91 9292 93 9494 95 9696 97 9898 99

100100 101 102102 103 104104 105 106106 107 108108 109

Sieving out the 3’s2 33 4 5 6 7 8 99

10 11 12 13 14 1515 16 17 18 19

20 2121 22 23 24 25 26 2727 28 29

30 31 32 3333 34 35 36 37 38 3939

40 41 42 43 44 4545 46 47 48 49

50 5151 52 53 54 55 56 5757 58 59

60 61 62 6363 64 65 66 67 68 6969

70 71 72 73 74 7575 76 77 78 79

80 8181 82 83 84 85 86 8787 88 89

90 91 92 9393 94 95 96 97 98 9999

100 101 102 103 104 105105 106 107 108 109

Sieving out the 5’s2 3 4 55 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 2525 26 27 28 29

30 31 32 33 34 3535 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 5555 56 57 58 59

60 61 62 63 64 6565 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 8585 86 87 88 89

90 91 92 93 94 9595 96 97 98 99

100 101 102 103 104 105 106 107 108 109

Sieving out the 7’s2 3 4 5 6 77 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 4949

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 7777 78 79

80 81 82 83 84 85 86 87 88 89

90 9191 92 93 94 95 96 97 98 99

100 101 102 103 104 105 106 107 108 109

Result of Sieving

• By the time I have sieved out the 7’s, a number that escapedescaped sieving would have to NOT be divisible by 2, 3, 5, or 7.

• UnUn-sieved numbers would therefore be divisible ONLY by 11 or numbers bigger than 11.

• The smallest such number (not counting 11 itself) is 11 × 11, which is 121.

• So . . .

Result of sieving (cont.)

• By sieving up to 7 (which is the 4th prime), I found all primes less than 1111.

In general:

• By sieving up to the nn-th prime, I can find all primes less than the square of the (nn+1)-th prime.

Topic 2

*

The Basel Problem

The Basel Problem — Preamble

Math allows you to add up an infinityan infinity of numbers and get a definite, finitefinite sum.

Example →So this infinity of numbers

adds up to one-third.

The trick is for the numbers to get smaller and smaller fast enoughfast enough.

0.3

0.03

0.003

0.0003

0.00003

0.000003

. . . . . .

+ ________________

0.3333333333333 . . .

Convergence and Divergence

0.3 0.03 0.003 0.0003 0.00003 0.000003 0.0000003 . . . . . . + ________________ 0.3333333333333 . . .

1 = 1 1/2 = 0.5 1/3 = 0.3333333333333 1/4 = 0.25 1/5 = 0.2 1/6 = 0.1666666666666 1/7 = 0.1428571428571 . . . . . . + ____________________

∞

The numbers in the yellow box get smaller very fast. You can keep adding them for ever: the sum is finitefinite.

The numbers in the pink box get smaller, but not fast enoughnot fast enough: If you add them for ever, the sum is infiniteinfinite.

The Basel Problem (cont.)

Consider the Harmonic Series:

(This was the pink boxthe pink box on the previous slide.)The numbers being added get smaller, but not fast enoughnot fast enough.

The sum is notnot finite. It “adds up to infinity.” The Harmonic Series divergesdiverges.This was proved by Nicole d’OresmeNicole d’Oresme around 1370.It was proved again by the BernoulliBernoulli brothers, Jakob (1682)

and Johann (1695). The brothers were successive Professors of Mathematics at the University of Basel, in Switzerland.

7

1

6

1

5

1

4

1

3

1

2

11

The Bernoulli Brothers

Jakob (1654-1705) Johann (1667-1748)

The Basel Problem (cont.)

What about this sum?

Does it also diverge? Or does it add to a finite number?

If the latter, what is the number?This was the Basel Problem. Both Bernoulli

brothers tackled it without success.It was solved at last in 1735 by Leonhard EulerLeonhard Euler, a

native of Basel (though he was living in St. Petersburg, Russia at the time).

222222 7

1

6

1

5

1

4

1

3

1

2

11

Leonhard Euler

• Swiss, 1707-1783• Ranked 11stst among

mathematicians in Murray’s Human Accomplishment.

• Studied under Johann Bernoulli at Basel.

• Prolific, pious (Calvinist), domestic.

• St. Petersburg Academy of Sciences, 1727-41 and 1766-83.

• Court of Frederick the Great, 1741-1766.

Charles Murray’s 2003 bookHuman AccomplishmentHuman Accomplishment

Mathematicians ranked

The Basel Problem – Solved!

Euler proved that

He then pursued the matter further and came up with:

The Golden KeyThe Golden Key

Topic 3

*

The Golden KeyThe Golden Key

From Particular to General

Fruitful math is often done by starting from a particularparticular result and generalizinggeneralizing it.

Having solved the Basel problem, Euler asked: “What if, instead of squaressquares in the denominators, I had some other power some other power ? What can we say about this infinite sum:

. . . where ‘ss’ is any number at all ?”

ssssss 7

1

6

1

5

1

4

1

3

1

2

11

When Does the Sum Exist?

Well, when ss = 1, this is just the Harmonic Series, which diverges, because the terms don’t get smaller fast enough.

When ss is less than 1, the terms get smaller even more slowly. In fact, when ss is less than zero, they don’t get smaller at all – they get bigger!

So when ss = 1 or less, the sum diverges “to infinity.”

When ss = 2 this is the Basel problem. Euler showed that it

then adds up to π2/6 .

In fact there is a definite sum for any value of ss greater than

1.

ssssss 7

1

6

1

5

1

4

1

3

1

2

11

The Zeta Function

So long as ss > 1, when I put in a number ss (the argument), I will get out a definite sum (the value).

This means I have a function. It is called the the zeta functionzeta function.

Here is its graph.

sss

s4

1

3

1

2

11

Euler’s great stroke of genius

Euler’s great stroke of genius was to apply to apply the sieve of Eratosthenes to the zeta the sieve of Eratosthenes to the zeta functionfunction.

Why not? In both cases you start off with a list of all the whole numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, . . .

You end up with only the prime numbers.

Like this . . .

Sieving out the 2’s

Here’s the zeta function. Call this “Expression 2a.”

Multiply both sides by 1/2s:

Call that “Expression 2b.”

Now subtract Expression 2b from Expression 2a:

Look! – I sieved out the 2’s!

ssssss

s7

1

6

1

5

1

4

1

3

1

2

11

ssssssss

s14

1

12

1

10

1

8

1

6

1

4

1

2

1

2

1

ssssssss

13

1

11

1

9

1

7

1

5

1

3

11

2

11

(A Slight Difference in the Sieving)

(Note a slight difference in my sieving technique.

Working the original Sieve of Eratosthenes, I left the first instancethe first instance of each prime standing.

Here I sieve out that first instance with all the rest.)


Call that last result “Expression 3a.”





ssssssss

13

1

11

1

9

1

7

1

5

1

3

11

2

11

ssssssss

33

1

27

1

21

1

15

1

9

1

2

11

3

1

ssssssss

17

1

13

1

11

1

7

1

5

11

2

11

3

11


Call that last result “Expression 5a.”





ssssssss

17

1

13

1

11

1

7

1

5

11

2

11

3

11

sssssssss

65

1

55

1

35

1

25

1

5

1

2

11

3

11

5

1

sssssssss

19

1

17

1

13

1

11

1

7

11

2

11

3

11

5

11

The Golden Key – At Last!

Just as with the original sieve, I can go on sieving for ever.

My final result will be:

Or, to rearrange slightly:

. . . With zeta on the left and all the primes represented on the right.

*** That is The Golden KeyThe Golden Key! ***

12

11

3

11

5

11

7

11

11

11

13

11

s

ssssss

ssssss

s

13

11

1

11

11

1

7

11

1

5

11

1

3

11

1

2

11

1

Why a “Key”? Why “Golden”?More formally:

The Greek capital pi says “Multiply together all such expressions, for all (in this case) primes pp.” (What Σ does for addition, Π does for multiplication.)

This is a key because it unlocks a door between Number Number TheoryTheory (“the higher arithmetic”) and Function TheoryFunction Theory (graphs, smoothness, calculus, etc.)

It is golden because all the power of function theory can now be brought to bear on problems about primes.

11 p

sps

Topic 4

*

Zeta’s Hidden Depths

What Use is Zeta?

The Golden Key tells us that all the properties of the all the properties of the prime numbers are prime numbers are contained somehow in contained somehow in the zeta functionthe zeta function..

But what use is that? Zeta looks really boring.

Remember its graph →Ah, but zeta has hidden

depths!

First, a slight detour.

A Slight Detour

Here is a different function, also defined by an infinite sum.

Like zeta, it exists only in a limited range.

If xx is 1 or more, the sum diverges “to infinity.”

Likewise if xx is -1 or less.

Graph of the function

54321 xxxxxxS

A Slight Detour (cont.)

However: If

Then

Subtracting

So

Well, that’s nice. But . . .

54321 xxxxxxS

65432 xxxxxxxSx

11 xSx

x

xS

1

1

A Slight Detour (cont.)

. . . S(x) and 1/(1-x) have different graphs.

The graphs are identical between -1 and +1,

but outside those bounds S(x) has no values (because the infinite sum diverges),

while 1/(1-x) has perfectly good values: e.g. 1/(1-3) = -½.

A Slight Detour – The Moral

The moral of the detour is:

• Using an infinite sum to express a function may “work” over only a partonly a part of the

function’s range.

• The function may have values even where the infinite sum “doesn’t work.”

The Hidden Zeta

Is that the case with zeta? Of course it is!

Here is a graph of zeta

between ss = –4 and ss = 1 →

Here’s a graph between

ss = –14 and ss = 0 →

A Common Point of Bafflement

• “OK, I understand the business with S(x). There’s that honest, well-defined function 1/(1-x). There’s also the infinite sum, but it only works over part of the function’s range. Got that.”

• “With zeta, though, the infinite sum is all we have. There’s nothing equivalent to that 1/(1-x). Wha? Huh?”

The Infinite Sum for ζ(s)

Point of bafflement:

The sum only ‘works’ when s is greater than 1.

So how can I talk about ζ(½) or ζ(-1)?

Relief for the Baffled

• There are ways to define zeta outside the range where the infinite sum works.

• Unfortunately they involve heavy-duty math.• Best known is the functional equation

. . . which gives ζ(1 – ss) in terms of ζ(ss), using standard mathematical functions (sine, factorial).

• So if you know the value of ζ(4) , you can work out a value for ζ(-3).

sss

s s !12

1sin21 1

More Relief for the Baffled

• There are other expressions for zeta outside the range where the infinite sum works, the range s > 1.

• For any positive whole number nn, for example, the following thing is true for any s bigger than –2nn:

dx

x

xxB

n

ns

k

ks

k

B

ss

nsn

n

k

k

1 1212

1

2

12

2

12

22

22

1

1

1

Topic 5

*

Zeros of a Function

Function Argument, Function Value

• Remember how functions work.• You put a number in (the argumentargument); you get a

number out (the function valuefunction value).• If you put argument 7 into the function

x2 – 1, you get out the function value 48.• If an argument produces function value zero,

that argument is a zero of the functiona zero of the function.• The zeros of the function x2 – 1 are –1 and 1.

Either argument will yield function value zero.

Zeros of the Zeta Function

• Recall the graph I showed a few frames ago, of the values of ζ between –14 and 0.

• The graph crosses the x-axis at several points.

• Its value then is zero.• I have found some I have found some

zeros of the zeta zeros of the zeta function!function!




Trivial and Nontrivial

• Yes, we have spotted some zeros of the zeta function.

• In fact ζ is zero for every negative even whole number argument: –2, –4, –6, –8, –10, –12, . . .

• Unfortunately these are trivial zeros, of no great interest.

• The Riemann Hypothesis concerns nontrivialnontrivial zeros.

• So . . . Where the heck are they?

Topic 6

*

Complex Numbers

Imaginary Numbers

• By the rule of signs, the square of a number must be positive.

• E.g. 2 2 = 4, –2 –2 = 4.• There’s no way to get “–4” by squaring something.• Negative numbers have no square roots!• By the late 16th century this restriction was holding up

mathematical progress.• Some bold mathematicians began to allow square roots

of negative numbers.• They figured that you only need a square root for –1.• Call it i. Then the square root of –4 is 2i.• Now every negative number has a square root.• The square root of a negative number is called an

imaginary number. Regular numbers are real numbers.

Complex Numbers

• If you multiply two imaginary numbers you get a real number: (–2i)(3i) = 6.

• If you add two imaginary numbers you get another imaginary number: (–2i)+(3i) = i.

• If you multiply an imaginary number by a real number, you get an imaginary number: (–2i)(3) = – 6i.

• If you add a real number to an imaginary number, they “don’t mix”: –2+3i.

• These “don’t mix” numbers are terrifically useful in math, though. We call them complex numberscomplex numbers.

• A complex number has a real part and an imaginary part. For –2+3i the real part is –2, the imaginary part is 3i.

Complex Arithmetic

• Arithmetic with complex numbers is very easy.

• You just have to remember that i 2 = –1.

• Example: multiply 1–i by –2+3i.

• By the ordinary rules of algebra:

(1–i) (–2+3i) = –2 + 3i + 2i – 3i 2

= –2 + 3i + 2i + 3

= 1 + 5i

Topic 7

*

Complex Functions

Complex FunctionsOnce complex numbers are allowed, why not use them as arguments for functions?

Example: Use the number –2+3i as an argument for the function x 2 – 1. The square of –2+3i, by ordinary algebra, is –5–12i. Subtract 1 for the function value –6–12i.

Function theory using complex numbers as arguments and function values – The Theory of Functions of a Complex The Theory of Functions of a Complex VariableVariable – is a rich and rewarding field of mathematics.

More Advanced Functions of a Complex Variable

More advanced mathematical functions can often be defined by power series – that is, infinite sums of powers.

The exponential function, for instance:

Complex Functions in History

Complex Function Theory was a huge growth point in early 19th-century math. It was sexy.

By 1850 there was a big, solid body of theory.

Bernhard Riemann used that theory to investigate ζ as a function of a complex variablefunction of a complex variable.

The powerful tools of complex function theory could then be applied to problems about prime numbers.

Bernhard Riemann

• German, 1826-1866• Tied 9th (with Pascal)

among mathematicians in Murray’s Human Accomplishment.

• Introverted, poor, ill, pious (Lutheran).

• Studied under Gauss at Göttingen.

• Brilliant imaginative mathematician.

Carl Friedrich Gauss

• German, 1777-1855• Ranked 4th among

mathematicians in Murray’s Human Accomplishment.

• Supervised Riemann’s doctoral thesis (1851).

• “Parva sed matura.”• First stated the Prime Prime

Number Theorem Number Theorem (1792)

Topic 8

*

The Distribution of Primes

The Prime Number Theorem (PNT)

If N is a whole number and π(N) is the number of prime numbers less than N, then π(N) approaches ever more closely to N / log(N) as N gets larger.

1,0001,000,000

1,000,000,0001,000,000,000,000

N

Number of primes

less than N

16878,498

50,847,53437,607,912,018

N / log(N)

14572,382

48,254,94236,191,206,825

Percentageerror

-16.05-8.45-5.37-3.91

To proveprove the PNT was a great challenge in 19th-century math.

The theorem was proved at last in 1896 by two French mathe-maticians, Jacques Hadamard and Charles de la Vallée Poussin.

Prime Numbers “Thin Out”

The PNT was the first good idea about the the distribution of primesdistribution of primes.

The prime numbers are scattered among the other numbers in a way that is at the same time randomrandom (you never know when the next one will appear) yet orderlyorderly (there are rules for how many you can expect to find). This is the fascination.

The main feature of the distribution of primes is that they “thin out.”they “thin out.”

In higher ranges of numbers, there are fewer primes.

The Primes “Thin Out” (cont.)Start with one thousand and count forward in blocks of 100. So the first block goes from 1,001 to 1,100. The second block goes from 1,101 to 1,200,… and so on.

The numbers of primes in the first ten blocks are: 16,12,14,11,17, 12, 15, 12, 12, 13 – average 13.4 per hundred numbers.

If, instead of starting with one thousand, you start with one million, then the numbers of primes in the first ten blocks are: 6, 10, 8, 8, 7, 7, 10, 5, 6, 8 – average 7.5 per hundred.

If, instead of starting with one thousand, you start with one trillion, the numbers are: 4, 6, 2, 4, 2, 4, 3, 5, 1, 6 – average 3.7.

This is the “thinning out” of the primes.

The PNT was the first mathematical expression for this observed phenomenon.

Topic 9

*

Riemann’s 1859 Paper

Riemann’s 1859 Paper• Title: “On the Number of Primes Less

Than a Given Quantity”

• Asks: How many primes are there between 1 and x x ?

• Begins with The Golden Key.

• Applies complex function theory to the ζ function.

• Arrives at an exact answer!

Riemann’s Answer

• How many primes are there between 1 and xx?

• Riemann’s answer:

where

n

n xJn

n

x ttt

dtxLixLixJ

log12log)(

2

Good Grief!

• If you’re not a seasoned mathematician, that looks scary.

• Yet in fact it’s made up of standard functions and methods: the Möbius μ-function, the log function, summation and integration.

• Only Li, the logarithmic-integral function, is not elementary. Here’s its graph for real arguments.

Note on the Log-integral Function

The log-integral function Li(x) is got by integrating 1/log(t) from t = 0 to t = x (“American” definition), or from t = 2 to t = x (“European” definition).

When t = 1, log(t)=0, so that 1/log(t) is infinite. This creates issues with the “American” definition!

The issues can be finessed, however; and Riemann used the “American” definition. Most math software packages also use it (e.g. Mathematica), so I use it too.

For any x, the two definitions differ by 1.04516378011749278… The “American” Li(x)

The Möbius μ-function

The Möbius μ-function is an arithmetic function.

That means that only positive whole-number arguments are allowed.

You are allowed to discuss μ(5) or μ(1000000000000), but you may NOT discuss μ(½) or μ(-3).

Definitionμ(n) has the value 1 when n = 1, and also when n is the product of an even number of different primes: for example 2170=2x5x7x31, so μ(2170)=1.

μ(n) has the value -1 when n is the product of an odd number of different primes: for example 561=3x11x17, so μ(561)=-1.

μ(n) is 0 otherwise: for example 20=2x2x5, so μ(20)=0.

The Möbius μ-function (cont.)N μ(N)

1 1

2 -1

3 -1

4 0

5 -1

6 1

7 -1

N μ(N)

8 0

9 0

10 1

11 -1

12 0

13 -1

14 1

N μ(N)

15 1

16 0

17 -1

18 0

19 -1

20 0

21 1

N μ(N)

22 1

23 -1

24 0

25 0

26 1

27 0

28 0

N μ(N)

29 -1

30 -1

31 -1

32 0

33 1

34 1

35 1

The Heart of the Matter

• The sensational part of Riemann’s result is

• This says: “For all possible values of ρ (Greek ‘rho’), add up Li(xρ).”

• What are these ρ? Why, they are the nontrivial zeros of the ζ function!

• They emerged from Riemann’s attack on The Golden Key via complex function theory.

• They are – no surprise – complex numbers.• They lie at the heart of Riemann’s topic: the distribution

of the prime numbers.

xLi

The Nontrivial Zeros

• There are infinitely many of them.• The first 20 are listed here →• Riemann himself computed the first

and third (not very accurately).• For each nontrivial zero a + bi there is

also one equal to a – bi.• J.P. Gram computed the first 15

nontrivial zeros in 1903.• Logician Alan Turing in 1953 computed

1,104 of the little devils.• In 2004, Gourdon and Demichel

computed ten trillion!• Every single one computed to date Every single one computed to date

has real part one-halfhas real part one-half.

0.5 + 14.134725142i0.5 + 21.022039639i0.5 + 25.010857580i0.5 + 30.424876126i0.5 + 32.935061588i0.5 + 37.586178159i0.5 + 40.918719012i0.5 + 43.327073281i0.5 + 48.005150881i0.5 + 49.773832478i0.5 + 52.970321478i0.5 + 56.446247697i0.5 + 59.347044003i0.5 + 60.831778525i0.5 + 65.112544048i0.5 + 67.079810529i0.5 + 69.546401711i0.5 + 72.067157674i0.5 + 75.704690699i0.5 + 77.144840069i . . . . . . . . . . . . . .




Riemann’s Statement of the Hypothesis

• Riemann guessed that all nontrivial zeros of the ζ function have real part one-half.

• In the 1859 paper “On the Number of Primes Less Than a Given Magnitude” he states an equivalent fact, then adds →

“One would of course like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting some fleeting vain attempts vain attempts (einigen flüchtigen vergeblichen Versuchen) because it is not necessary for the immediate objective of my investigation.”

150 Years On

• We still do not know whether all nontrivial zeros of the zeta function have real part one-half.

• There is now a great mass of theory about prime numbers.

• A large part of it consists of theorems whose statement begins: “Assuming the truth of the Riemann Hypothesis . . .”

• A resolution of the RH – a proof that it is true, or a proof that it is false – would revolutionize this branch of mathematics.

Footnote 1

*

The Chebyshev Bias

Pafnuty Lvovich Chebyshev

• Russian (1821-94)• Born to a prosperous

military family (his father fought against Napoleon).

• Studied at Moscow University.

• Taught at St. Petersburg.• Did important work on

prime numbers.• Came close to proving

the PNT (1850).

The Chebyshev Bias -- ExamplesDivide a prime number (other than 2) by 4. The remainder must be either 1 or 3.

For the primes up to 101 (which are: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101) the remainders are 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 3, 1, 3, 3, 1,3, 3, 1, 1, 1. That’s 12 remainder-1 primes and 13 remainder-3 primes – a slight biasbias in favor of remainder-3 primes.

For the primes up to 1,009 the counts are 81 and 87 – a stronger biasbias in favor of remainder-3s.

Up to 10,007 the counts are 609 and 620 – still favoring remainder-3s!

This biasbias is only violated – and very briefly – at 26,861, when reminder-1 primes take the lead.

In the first 5.8 million primes, remainder-1 primes hold the lead at only 1,939 places.

This is a Chebyshev biasChebyshev bias.

The Chebyshev Bias (cont.)

Instead of dividing by 4, divide by 3. Now the remainder (ignore p = 3) is either 1 or 2.

The bias is to 2. This bias is not violated until p = 608,981,813,029! (Discovered in 1978.)

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Chebyshev biases express deep properties of the whole numbers.

Even though you may wait a long time for the first violation of a Chebyshev bias, each bias is violated infinitely many times.

Chebyshev Biases (cont.)

Chebyshev biases give us a good informalinformal reason to think that the RH may not be true.

Similar – though mathematically deeper – biases are found all over prime number theory.

The pattern is: A certain theorem (“Remainder-3 primes will always outnumber remainder-1 primes”) is true for all the numbers we check, into the thousands or even millions.

Then, at some very high value, a violation occurs!

The RH has been found true for trillions of nontrivial zeroes, butbut…….

Footnote 2

*

The Riemann Hypothesis Song

The Riemann Hypothesis Song

The Riemann Hypothesis Song(continued)

The Riemann Hypothesis Song(concluded)

Documents

Introduction