(1)

a

a

d|n

a

a

n<y

a

a

a

a

a

1+a

x

1

a

2

1

3/2+a

2

1

−1

−2

2

−1

2

2

a

2πinh/k

x

1

a

2

1

3/2+a

5/4+a/2+ε

[351]

(2)

0

0

x

1

0

2

0

3/2

5

0

4

2

4

(1−a)/2

T

1

2

a

1/4+a/2

a

1/4+a/2

−1/2

a

a

a

(1+a)/2

a

(1+a)/3+ε

a

a

a

(1−a)/2

3/4+3a/2

(3)

a

a

a

−1/4+a/2

−1/2

a

1/4+a/2

2

1

n≤uX

a

−3/4−a/2

a

n≤Z

a

2

1

uX

−1

a

a

a

ε

a

a

a

a

2

a

2

a

a

a

2

−1/4

a

a

−1/2

a

a

a

a

−1/4

a

2

a

2

a

a

a

a

a

2

−1/4

a

a

−1/2

x

x/2

a

2

1

2

3

30

3

1/4

1

3

1/2

1

x

x/2

a

2

2

x

x/2

a

a

(4)

3

x

x/2

a

2

30

x

x/2

a

2

1

1

3/2+a

3/2+a

2

2

3

30

a

1

10

11

10

2

x

x/2

1/2+a

n≤8x

2

2

11

2

x

x/2

1/2+a

m,n≤8x m6=n

a

−3/4−a/2

2

max(1,n/(4x))

11

±

±

x

x/2

1/2+a

m,n≤8x m6=n

4πi(

√m±√ n)√

y

1

2

±

1+a

ξ2

ξ1

−1/2

m,n≤8x m6=n

4πi(

√m±√ n)√

y

1+a

2

j=1

m,n≤8x m6=n

−1

4πi(

√m±√ n)

ξj

+

(5)

m6=n

−1/2

1+a

n≤8x

2

1/2

1+a

n≤8x

a

2

−a−1

11

10

2

n≤8x

2

x

x/2

1/2+a

1+a

10

2

3/2+a

3/2+a

n≤8x

2

1+a

n≤8x

2

n≤x

a

2

−3/2−a

−1/2−a

−1/2−a

10

n≤x

a

2

−3/2−a

2

−1

−1/2−a

10

1

3/2+a

3/2+a

n≤x

a

2

−1

2

−1

2

10

2

2

3/4+a/2

x

ξ

−3/4−a/2

a

a

2

3/4+a/2

m≤2x

a

−3/4−a/2

n≤2x

a

(6)

x

ξ

−1/2

2

1

uX

−1

n≤2x

a

−1/2

y∈{ξ,x}

−1

uY

uX

−1

−4πi

√nt x

ξ

−1/2

4πi(

√t±√ m)√

y

y∈{ξ,x}

uY

uX

−1

−1

−4πi(

√nt−(√ t±√

m)√ y)

y∈{ξ,x}

uY

uX

−1

−1

4πi(

√y−√ n)√

t

y∈{ξ,x}

−1/2

−1

−1

Y →∞

2

1

uX

−1

−1

−2

(7)

3

x

x/2

n≤2x

a

−2

2

x

x/2

n≤2x

a

2

−2

n≤2x

a

2

x

x/2

−2

n≤2x

a

2

30

z

z,p

−∞

−π(pv)2

1/2−z

z

2

z,p

z

1+z

1+z

1+z

z

(1+z)/2

n=1

z

−(1+z)/2

z

z,p

(1+z)/2

n=1

z

−(1+z)/2

−∞

−π(pv)2

3/2

z

2

(8)

−1/4+z/2

n=1

z

−5/4−z/2

−∞

−π(pv)2

z

2

z

2

−∞

Av−Bv2

A2/(4B)

−∞

−π(pv)2+4πivny

−4πny/p2

−3/2

−p

−∞

−π(pv)2+4πiv

√ny

1/2

−2

−4πny/p2

−1

−p

z,p

z,p

z,p

−3/4+z/2

3/4

z

z

−1/2

z,p

z,p

z,p

z,p

−3/4+z/2

a,p

a,p

(1)a,p

(2)a,p

−3/4+a/2

(1)a,p

1/4+a/2

n=1

a

−3/4−a/2

−4πny/p2

(9)

(2)a,p

−1/4+a/2

n=1

a

3

−1

2

−5/4−a/2

−4πny/p2

3

2

2

3

z

3/4

z

2

1 uY

uX

−l

−t/V +ik

√t

−l

−2

−l

−2

−l

−2

y

0

a

4

3/4+a/2

n=1

a

−5/4−a/2

−3/4+a/2

2

ν=0

ν

−ν/2

0

2

1

2

4

y

0

a

3/4+a/2

1/2

3/4+a/2

(10)

p

−3/4−a/2

−4πty/p2

n>uX

a

p

uX

p0

a

a

uX

p0

1+a

t v=uX

uX

p0

a

a

uX

p

a

p

a

uX

p00

t

uX

a

2

1

n>uX

a

p

1

2

Y →∞

3

1

2

1

p

a

2

2

1

uX

p

a

3

2

1 uY

uX

p00

t

uX

a

1

−1/2

2

−1/2

(11)

2

2

−1

−3/4−a/2

1

1

−1

p

t

0

a

2X

t=X

−1

2X

X

p0

t

0

a

−1

11

−1

12

11

−1/4−a/2

12

n=1

−3/4

2X

X

p0

3/4+a/2

2X

X

p0

−3/4+a/2

2X

X

p0

121

122

123

p0

−3/4−a/2

1/2

−1/2

5

−2

6

−1

−4πty/p2

5

6

2X

X

p0

3/4+a/2

1/2

1/2

−1

121

1/4

1/2

−1/4

122

1/2

−1

123

−3/4−a/2

3

3

n=1

a

−5/4−a/2

2

1 uY

uX

p00

3/4+a/2

ty=uX

−3/4+a/2

2Y

X

p00

−3/2

p00

−7/4−a/2

2

1

p0

uY

uX

a

−3/4+a/2

y

2

−Y y/p2

(12)

3

n=1

a

−5/4−a/2

2

1 uY

uX

p0

3/4+a/2

0

−3/2

y

2

−Y y/p2

3/4+a/2

0

3/4+a/2

−1

1/2

−1/2

7

−1

−1/2

−3/2

7

2

2

1 uY

uX

p0

3/4+a/2

0

1/2

1/2

−3/2

−2

1/2

−1

−1/2

2

1 uY

uX

−1

−4πyt/p2

−1

3

1/2

n≤Z

a

−3/4−a/2

y

2

−Y y/p2

1/2

−1

−3/4−a/2+ε

−1/2

(1)a,p

1/4+a/2

2

1

n≤uX

a

p

3/4+a/2

n≤Z

a

−3/4−a/2

Y →∞

−1/4+a/2

(13)

(2)a,p

3

(2)a,p

−1/4+a/2

−1/2

p→∞

a,p

a

−∞

−π(pv)2

1/2−a

−1

a,p

a

−∞

−π(pv)2

1/2−a

a

2

a

a

2/3

−2/3

−π(pv)2

a

2

a

−1

2/3

0

−π(pv)2

|n−y|≤2yv

a

−1

2p/3

0

−πv2

|n−y|≤2yv/p

a

−1

a

a

3/4+a/2

n≤Z

a

−3/4−a/2

p→∞

Y →∞

−1/4+a/2

−1/2

p→∞

Y →∞

2

1

uX

−1

(14)

−3/4−a/2

−3/4−a/2

−1/4+a/2

a