• Nie Znaleziono Wyników

# On values of

N/A
N/A
Protected

Share "On values of"

Copied!
34
0
0

Pełen tekst

(1)

2

[359]

(2)

1

2

0

2

0

0

1

−k

a bc d

az+bcz+d

k,χ0

0

0

1

0

0

0

k

0

1

n=0

n

l

l

n=0

ln

k

k,χ0

0k,χ0

∞,0k,χ

0

k,χ0

k,χ0

∞,0k,χ

0

∞,0k,χ0

∞,0k,χ

0

k,χ0

0

00

−1

−n0

i

00

00

(0)0

n0

n=0

−n

n

(0)0

0

0

n0

∞,0k,χ

0

(0)0

z→∞

−k

(0)0

k,χ0

0k,χ0

n0

n=0

−n

n

00

(0)0

n0

n=1

−n

n

0k,χ0

(3)

0

−1

−n0

n0

n=0

−n

n

∞,0k,χ

0

0

0

∞,0k,χ

0

k,χ0

k

k,χ0

k,χ0

0k,χ0

0

0

0

0

0

2

2

n=−n0

n

0

−10

(i/M )0

k,χ0

n=0

n

(i/M )0

M,i

2

(i/M )0

(i/M )0

n0

n=0

−n

n

0

0

−1

n

(0)0

0

(0)0

−1

−n0

k,χ0

(0)0

0

−1

−n0

0k,χ0

p

k

k

∞,0k

∞,0k

p

p

k

k

p

∞,0k

∞,0k

p

k−1

(0)0

(0)0

p

k

k

p

0k

0k

0

00

−1

−n0

k

0

k−1

00

−1

−n0

k

(4)

k

k

p|N

0

−1

−n0

0k

0

−1

−n0

0k

0k

0k

p

k,χ0

k,χ0

K

K

N

N

N

N

N

ψ

ψ

M

M

M

g

1

g

g

g

i=1

(i)

i

(i)

(1)

(g)

0

0

0

−1K

0

−1

−1K

g

k,A

0

0

0

k

0

γ,δ

−k

−s

s=0

0

0

−1K

0

−1K

NN0

NN0

N

0

N0

0

NN0

ψk,ψ0

k

g

K−1/2

−1

N

NN0

−1

A∈CN

0

0

−1

K

0

k,A

0

0

0

(5)

0

0

0

K

2

0

0

ψk,ψ0

ψk,ψ0

g

ν∈d−1K , ν0

O⊃B⊃νdK

0

−1

K

k−1

K

0

0

k,ψ

ψk0

k,ψ

ψk0

α βγ δ

2

N

k,ψ

k−1

P|N P-(γ,N)

−1

K

(γ,N)

ψ

−g

−1/2K

K

ψk

k

g

Kk−1/2

K

−1

P|N P-(γ,N)

−1

K

(γ,N)

ψ

ψgk,ψ0

ψk,ψ0

0

0

0

0

ψgk,ψ0

gk,χ0

ψgk,ψ0

g

n=1

ψk−1,ψ0

(6)

ψk−1,ψ0

## (n) := X

ν∈d−1K , ν0 tr(ν)=n

O⊃A⊃νdK

0

−1

K

k−1

0

ψk−1,ψ0

k−1,ψ

ψk−10

gk,ψ

k,ψ

ψgk

ψk

0

0

gk,ψ

ψgk

gk,χ0

gk,ψ

0

−1

P|N P-(M,N)

−1

K

(M,N)

ψ

−g

K1/2

0

K

ψgk

k

g

k−1/2K

0

K

ψ

gk,ψ

gk,χ0

ψgk

0gk,χ0

g,ψ

∞,0g,χ0

K

−1/2

K

µ∈O/N, µ0

−g

K

−g

−1/2

0

0

(7)

0

K

g

n0

n=1

−n

k−1,ψ

0

−1

−n0

gk,χ0

0

−1

−n0

0g,χ0

K

0

−g

−1/2

00

K

g

n0

n=1

−n

0,ψ

0

00

−1

−n0

g,χ0

00

K

k

g

−k+1/2K

n0

n=1

−n

ψk−1

00

−1

−n0

gk,χ0

00

−1

−n0

0g,χ0

O

g/2

K

0

P0

g,ψ0

K

g

n=1

0,ψ,P

0,ψ,P

## (n) := X

ν∈d−1K , ν0 tr(ν)=n

O⊃A⊃νdK (A,P)=O

g

0

−1

−n0

0g

0

K

g

−1

n0

n=1

−n

0,ψ,P

0

(8)

K

K

0

K

g

K

−1

n0

n=1

−n

0,ψ

0

−n0

0g

K

K

0

−1

00

K

g

−1

n0

n=1

−n

0,ψ,P

0

00

−1

−n0

g

0

−1

00

K

g

−1

−1

00

−10

m0

n=1

−n

0,ψ

n0

n=1

−n

0,ψ,P

0

m0

0g

0

0

00

−1

−n0

g

g,ψ

g

n=1

k−1,ψ

K

K

K

g

g,ψ0

g

−1

K

K

−1

K

−1

K

g,ψ0

−1

K

−1

g

−10

m0

n=1

−n

0,ψ

0

K

00

−1

K

−1

g

n0

n=1

−n

0,ψ,P

D

(9)

K

F

K

F

K

0

0

0

−g

−1/2

00

g−1

K

−1F

n0

n=1

−n

0,ψ

0

00

−1

−n0

g,χ0

0

g−2

K

−1F

n0

n=1

−n

0,ψ

0

−n0

0g

0

−1

00

g−1

K

−1F

−1

n0

n=1

−n

0,ψ,P

0

00

−1

−n0

g

K

0

ψk−1,ψ0

## (n) = X

|m|<n√ DK m≡nDK(mod 2)

## X

O⊃A⊃((m+n√ DK)/2)

0

K

−1

k−1

O

0,ψ,P

## (n) = −ψ(P) X

|m|<n√ DK m≡nDK(mod 2)

## X

O⊃A⊃P−1((m+n√ DK)/2)

−1K

K

K

K

K

(10)

2,ψ

2

0,ψ

0,ψ,P

0,ψ

0,ψ,P

## (n)) = − X

ν∈d−1K , ν0 tr(ν)=n

P⊃A⊃νdK

O

O

7

7

O

i

i

7

i

02

K

i

−1

|m|<√ 79

## X

O⊃A⊃P−17 (m+√ 79 )

i

7

7

2

7

i

7

K

i

−1

2

2

2

K

1

K

5

K

3

O

7

k−1,ψ

k−1

k−1

k−1

k−1

k−1

k−1

5

k−1

k−1

k−1

k−1

k−1

k−1

k−1

k−1

2

4

k−1

k−1

k−1

k−1

k−1

k−1

k−1

k−1

k−1

k−1

k−1

k−1

3

4

6

8

10

K

2

K

4

K

0

K

K

1

K

5

K

0

K

K

2

(11)

K

4

K

0

K

K

1

K

5

K

3

F/K

F/K

F

F

K

F

K

F

K

F

−g+1

F/K

D

−νA

K

K

2

0

0

0

0

−νP

K

np

Nm(ν)p

D

2

K

|m|<√ DK

m≡DK(mod 2)

O⊃A⊃((m+√ DK)/2)

K

32

2

0

0

2p

0

8

2,χ8

K

F

−1/2

−1

0,ψ

K

(12)

|m|<√ 2

A⊃(m+√ 2 )

2

32

2

32

|m|<√ 17 m odd

A⊃((m+√ 17 )/2)

2

|m|<√ 7

A⊃(m+√ 7 )

K

K

K

0

0

0

13

13

13

13

13

13

0

13

2,χ0

## X

|m|<√ DK m≡DK(mod 2)

O⊃A⊃((m+√ DK)/2)

(13)

|m|≤3 m odd

A⊃((m+√ 13 )/2)

K

K

2

2

K

2

2

|m|≤3 m odd

A⊃((m+√ 17 )/2)

2

2

2

2

2

2

9+229

9−29

2

16

16

F/K

F/K

×

2

0

−4

−4

(n−1)/2

3,χ−4

(14)

−1/2

−1

0,ψ

## X

ν∈d−1K , ν0 tr(ν)=1

O⊃A⊃νdK

3

2

2

2

2

3

03

5

05

7

07

i

0i

−1K

K

3

5

7

03

3

K

5

K

7

K

03

K

P−10

3

K

9

0

0

0

0

3

K

5

K

7

K

03

K

3

2

3

5

7

03

9

0

04

F/K

0,ψ

K

F/K

−1K

(15)

K

22

23

25

025

19

019

0019

00019

2

25

025

2

25

O

2

2

25

2

3

D

0

4

F/K

0,ψ

0

4

4

F/K

4

−1/2

−1

0,ψ

0,ψ

4

3

2

2

3

3

03

7

07

3

7

03

07

−1K

0

2

19571

3

2

−1K

K

3

7

F/K

14

−13

−17

3

2

F/K

3

K

7

K

F/K

0

3

K

03

K

## = (−1) · (−1) = 1

Cytaty

Powiązane dokumenty

Abstract: Using the technique associated with measure of non- compactness we prove the existence of monotonic solutions of a class of quadratic integral equation of Volterra type in

L e, Upper bounds for class numbers of real quadratic fields,

(Note that h/n divides H because the Hilbert class field of k is an abelian unramified extension of K of degree h/n.) Since examples with H = h/n do exist, this index, for a

Given the many similarities between function fields and number fields it is reasonable to conjecture that the same is true of real quadratic function fields (see, for example,

These ideals (of the integral group algebra of the Galois group) annihilate the ideal class group of the field and, for non- real fields, their indices give interpretations of the

(One can also give a different proof by adapting that of Proposition 3 below; see the remark following that proposition.).. In this paper we obtain several new estimates of l(q) and

To prove the above theorems we need some general results from the theory of diophantine equations, moreover, several proper- ties of Stirling numbers.. Let K be a finite extension of

classes of primitive hyperbolic transformations of PSL(2, Z). The set of reduced quadratic numbers equivalent to x will be denoted by x.. where τ runs through the set of classes.