Transcendental Number Theory: the State of the Art

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KIAS

the State of the Art

Michel Waldschmidt

http://www.math.jussieu.fr/∼miw/

(2)

(3)

n≥1

−n!

(4)

d

−d

(5)

n≥1

−n!

N!

N

n=1

N!−n!

−n!

−(N+1)!

(6)

0

n≥1

−n2

N2

N

n=1

N2−n2

0

n≥N+1

−n2

−(N+1)2

2N+1

0

0

0

0

0

(7)

0

n≥1

−n2

(8)

(9)

(10)

(11)

(12)

β

(13)

2

3

1

n

β1

βn

(14)

1

2

1

2

(15)

1

2

1

2

β

(16)

1

2

1

2

β

1

2

1

2

1

2

1

2

β

1

2

β

2

1

(17)

1

2

1

2

β

2

π

(18)

z

(19)

π

πz

1

2

1

2

1

2

1

2

(20)

1

n

1

n

1

n

1

n

1

n

1

n

1

n

1

n

x1

xn

(21)

π2

log 3

log 3

π

e

e

(22)

n

(23)

2x2≤1

x2+y2≤1

1<x<2

1>t1>t2>0

1

1

2

2

2

(24)

b

a

b

a

b

a

b

a

c

a

b

c

ϕ(b)

ϕ(a)

b

a

0

(25)

b

a

0

(26)

x2+y2≤1

1

−1

2

1

−1

2

−∞

2

(27)

1

0

3

(28)

γ

(29)

1

a−1

b−1

(30)

q

(31)

(32)

Yu.V. Nesterenko (1996) – Modular functions and transcendence questions

P(q) = E2(q) = 1 − 24

X

n=1

nqn 1 − qn, Q(q) = E4(q) = 1 + 240

X

n=1

n3qn 1 − qn, R(q) = E6(q) = 1 − 504

X

n=1

n5qn 1 − qn·

(33)

2

3

2

1

1

2

2iπτ

23

23

32

3

2

3

n

24

(34)

2

3

2

3

−2π

1

2

−2π

−2π

1

4

−2π

−2π

6

1

12

(35)

2

3

2

3

−π

3

1

3

4/3

−π

3

−π

3

6

12

(36)

π

π

3

(37)

2iπτ

(38)

(39)

(40)

(41)

5

5

(42)

5

5

π

5

(43)

n−1

k=0

(n−1)/2

−na+(1/2)

(44)

N−1

i=0

(1/2)−N z

(45)

×

×

×

×

×

×

N−1

i=0

(46)

(47)

b/2

a∈Q

ma

a

(48)

b/2

a∈Q

ma

a

(49)

b/2

a∈Q

ma

a

(50)

n→∞

(51)

n→∞

(52)

n→∞

(53)

n→∞

(54)

b0

b11

bmm

0

i

(55)

p

−s

−1

s→1

(56)

k

2

2

n

n

k=2

k

n→∞

n

0

n≥0

n

−s

(57)

0

n≥0

n+1

(58)

n≥0 Q(n)6=0

(59)

n=1

n=0

(60)

n=1

n=0

n=0

n=0

(61)

n=0

n=0

n=1

2

2

n=0

2

π

−π

π

−π

(62)

n=0

n≥1

s

(63)

n≥1

s

−2k

(64)

(65)

(66)

(67)

(68)

(69)

(70)

(71)

X

n1≥1

n−s1 1 X

n2≥1

n2−s2 = X

n1>n2≥1

n−s1 1n2−s2 + X

n2>n1≥1

n−s2 2n1−s1 + X

n≥1

n−s1−s2

1

2

1

2

1

2

2

1

1

2

−s −s

(72)

1

k

1

1

k

n1>n2>···>nk≥1

s11

skk

(73)

1

k

1

1

k

n1>n2>···>nk≥1

s11

skk

(74)

1

k

1

1

k

n1>n2>···>nk≥1

s11

skk

(75)

(76)

(77)

1>t1>t2>t3>0

1

1

2

2

3

3

(78)

1>t1>t2>t3>0

1

1

2

2

3

3

1>t1>t2>t3>0

1

1

2

2

3

3

(79)

1>t1>t2>t3>0

1

1

2

2

3

3

1>t1>t2>t3>0

1

1

2

2

3

3

(80)

p

1

k

0

1

p

p

p

p−2

p−3

0

1

2

(81)

p≥0

p

p

2

3

p

1

k

1

k

i

(82)

p

p−2

p−3

0

1

p

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