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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
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