Dramatis Personae

Robin Whitty

Queen Mary University of London

Riemann’s Hypothesis, Rewley House, 15 September 2013

Robin Whitty Dramatis Personae

Symbols

Z = {. . . ,−3,−2,−1, 0, 1, 2, 3, 4, . . .} the integers

R = the real numbers — integers, fractions, irrationals

C = the complex numbers — more later

e = 2.7182818284... Euler’s number

γ = 0.5772156649... EulerMascheroni (is it irrational??)

τ = 6.2831853071... circle constant (= 2π)

∞ = the size of Z (for our purposes)

i =√−1 imaginary — more later

Robin Whitty Dramatis Personae

Powers

20 = 1

21 = 2

22 × 23 = 22+3 = 25 = 32(

22)3

= 22×3 = 26 = 64( 6= 223= 28 = 256)

2−1 =1

2

21/2 =√2 = 1.41421356237...

eτ = 535.4916555247...

(eτ )√−1 = eτ×

√−1 = eτ i = 1

A cautionary tale:

1 =√1 =

√−1×−1 =

√−1×

√−1 =

(√−1

)2= −1

Robin Whitty Dramatis Personae

Capital Sigma Σ

1 + 2 + 3 + . . .+ n abbr.

n∑

k=1

k =1

2n(n + 1)

12 + 22 + 32 + . . .+ n2 abbr.n

∑

k=1

k2 =1

6n(n + 1)(2n + 1)

1r + 2r + 3r + . . .+ nr abbr.n

∑

k=1

k r = Sr (n)

r1 + r2 + r3 + . . .+ rn abbr.n

∑

k=1

rk =r (1− rn)

1− r

(sum of geometric progression)

1− 1

2+

1

3− 1

4+ . . . abbr.

∞∑

k=1

(−1)k+1

k= 0.6931471805...

Robin Whitty Dramatis Personae

Capital Pi Π

1× 2× 3× . . .× n abbr.n∏

k=1

k = n! (n factorial)

0! = 1! = 1, 2! = 2, 3! = 6, 4! = 24, . . .

12 × 22 × 32 × . . .× n2 abbr.n∏

k=1

k2 = (n!)2

21 × 22 × 23 × . . .× 2n abbr.n∏

k=1

2k = 21+2+3+...+n

= 2∑

n

k=1 k = 212n(n+1) =

((√2)n)n+1

22

1.3.42

3.5.62

5.7. . . abbr.

∞∏

k=1

(2k)2

(2k − 1)(2k + 1)=

1

4τ

(Wallis’s Product)

Robin Whitty Dramatis Personae

Pascal’s Triangular Cornucopia

(

n

k

)

or nCk

(

00

)

= 1,(

72

)

=(

75

)

= 21, etc

Robin Whitty Dramatis Personae

The Power of Pascal IRecall Sr (n) denotes the sum 1r + 2r + . . . nr =

∑

n

k=1 kr . In the

early 18th century, Jakob Bernoulli and Sansei Takekazu-KowaSeki independently discovered:

Sr (n) =1

r + 1

r∑

k=0

(−1)k(

r + 1

k

)

Bknr+1−k

where

B0 = 1, B1 = −1

2, B2 =

1

6, B3 = 0, B4 = − 1

30, B5 = 0, . . ..

E.g. S2(n) =1

2 + 1

((

3

0

)

B0n3 −

(

3

1

)

B1n2 +

(

3

2

)

B2n1

)

=1

3

(

1× n3 − 3×−1

2× n2 + 3× 1

6× n

)

=1

6

(

2n3 + 3n2 + n)

=1

6n(n + 1)(2n + 1)

Robin Whitty Dramatis Personae

The Power of Pascal IIThe Bernoulli numbers:

B0 = 1, B1 = −1

2, B2 =

1

6, B3 = 0, B4 = − 1

30, B5 = 0, B6 =

1

42, B7 = 0, . . .

How can we find B100? Are the odd B ’s (except B1) always zero?

Pascal to the rescue:

n∑

k=0

(

n + 1

k

)

Bk = 0, for n ≥ 1

which means if we know B0,B1, . . .Bn then we can get Bn+1.

E.g.

8∑

k=0

(

9

k

)

Bk =

(

9

0

)

B0 +

(

9

1

)

B1 +

(

9

2

)

B2 +

(

9

3

)

B3 +

(

9

4

)

B4 +

(

9

5

)

B5 +

(

9

6

)

B6 +

(

9

7

)

B7 +

(

9

8

)

B8

=

(

9

0

)

× 1 +

(

9

1

)

× −

1

2+

(

9

2

)

×

1

6+

(

9

4

)

× −

1

30+

(

9

6

)

×

1

42+

(

9

8

)

B8

= 1 + 9 × −

1

2+ 36 ×

1

6+ 126 × −

1

30+ 84 ×

1

42+ 9B8 = 0

Solve for B8: B8 =1

9

(

−1 +9

2−

36

6+

126

30−

84

42

)

= −

1

30

Robin Whitty Dramatis Personae

‘Convergent to’

Roughly, a sequence of numbers a1, a2, a3, . . . converges to limit Lif the difference between ak and L gets progressively closer tozero.

E.g. (1) the sequence of fractions1

1,2

1,3

2,5

3,8

5,13

8, . . ., consisting

of ratios of successive Fibonacci numbers, converges to the goldenratio ϕ = 1

2

(

1 +√5)

= 1.6180339887.... For short we can write

limn→∞

Fn+1

Fn= ϕ.

(2) the sequence of fractions1

1,3

2,7

5,17

12,41

29, . . ., defined by

a1 =n1

d1=

1

1, a2 =

n2

d2=

n1 + 2d1n1 + d1

, a3 =n3

d3=

n2 + 2d2n2 + d2

, . . . ,

converges to limit√2. That is, lim

n→∞an =

√2.

Robin Whitty Dramatis Personae

‘Asymptotic to’Roughly, a sequence of numbers a1, a2, a3, . . . is asymptotic to M

if the ratio of ak to M gets progressively closer to one (i.e. %error goes to zero).

This may involve two sequences a1, a2, a3, . . . , and b1, b2, b3, . . ..Their difference may get bigger and bigger: lim

n→∞|an − bn| = ∞.

But this still allows their ratios to converge: limn→∞

an

bn= 1. We

write an ∼ bn, for short (“an is asymptotic to bn”).

E.g. The factorial function n! is asymptotic to√τnnne−n.

n! ∼ √τnnne−n: the upper red

line plots n! (extended continu-ously); the lower blue line is Stir-ling’s approximation

Robin Whitty Dramatis Personae

An asymptotic for the Bernoulli numbers

The sequence of even Bernoulli numbers B0,B2,B4, . . . is

asymptotic, in absolute value, to the sequence2(2k)!

τ2k.

(−1)k+1B2k ∼ 2(2k)!

τ2k

B2k (thin red line) plotted against 2(2k)!

τ2k (wide blue line) in the range 11 ≤ n ≤ 20.

Plots of B2k −2(2k)!

τ2k for k = 11, . . . , 20 (note difference in vertical scales!)

Robin Whitty Dramatis Personae

The (natural) logarithm

What power of 2 gives me 16? The answer is written as afunction: log2(16) “log to base 2 of 16”. The function hasvalue 4 at argument 16 because 16 = 24.

What power of e = 2.71828... gives me 16? The answer isloge(16). This is calculated as ln(16) on your calculator (‘ln’ for‘natural log’) but mathematicians write it as log(16) (base e isassumed unless otherwise stated). The value of log(16) is2.77258... meaning (2.71828...)2.77258... = 16.

Plots of loga(x) for a = 2, e and10 (green, red and blue, respec-tively). For base e the slope asthe curve passes the horizontalaxis is precisely 1.

Rules: log ((a× b)c) = c log(ab) = c(log a+ log b) = log ac + log bc .

Robin Whitty Dramatis Personae

How to calculate logsOn your calculator, every ‘scientific function’ is calculated byadding enough terms in a suitable series:

For natural logs, the series is specified as

log(1− x) = −x − x2

2− x3

3− x4

4− . . .

This is valid provided −1 ≤ x < 1.

E.g. log 2 = log (1− (−1))

= −(−1)− (−1)2

2− (−1)3

3− (−1)4

4− . . .

= 1− 1

2+

1

3− 1

4+ = 0.6931471805...

Robin Whitty Dramatis Personae

The area integral

“The area between a curve y = f (x) and the horizontal axis

between x = a and x = b” is abbreviated to

∫

b

a

f (x)dx .

∫ 10 (− log x)dx = 1∫ 10 (− log x)2dx = 2

∫ 10 (− log x)3dx = 6∫ 10 (− log x)4dx = 24

Robin Whitty Dramatis Personae

Our star turn: the primes

Euclid’s Book IX, Proposition 20 They form an infinity.The Fundamental Theorem of Arithmetic They construct anypositive integer uniquely: n =

∏

pa11 pa22 · · · parr (r , primes pi andpowers ai determined uniquely n).They are unpredictable: 11 is prime, 1111111111111111111 is prime,

i.e.∑18

k=0 10k . What about

∑1010

k=0 10k?

Robin Whitty Dramatis Personae

A digression...

In a single week in April 2013 two massive breakthroughs inanalytic number theory were announced:

Week Goldbach: any odd number greater than7 is a sum of three odd primes (Harald Helfgott).E.g. 35 = 11 + 11 + 13.Previously known for all large odd numbers and(Tao, 2012) for five primes.

Small prime gaps: there are infinitely manypairs of primes separated by 70 million or less(Yitang Zhang). Quickly improved to approx.5000 or less.Previous best: smallest prime gap growsslower than primes themselves (Goldston–Pintz–Yıldırım, 2005).

Robin Whitty Dramatis Personae