Ekkehard Kr¨ atzel

(1)

Lattice points in some special three-dimensional

convex bodies with points of Gaussian

curvature zero at the boundary

Ekkehard Kr¨ atzel

Abstract. We investigate the number of lattice points in special three-dimensional convex bodies. They are called convex bodies of pseudo revolution, because we have in one special case a body of revolution and in another case even a super sphere. These bodies have lines at the boundary, where all points have Gaussian curvature zero. We consider the influence of these points to the lattice rest in the asymptotic representation of the number of lattice points.

Keywords: convex bodies, lattice points, points with Gaussian curvature zero Classification: 11P21, 11H06

1. Introduction and statement ofresult

Let

F

^denote^the^distance ^function^of^the^convex^body^PR

3

^. ^That ^is

F

⁽

t 1 , t 2 , t 3

⁾⁼

n

(

|t 1 | ^κ

⁺

|t 2 | ^κ

⁾

^k ^κ

⁺

|t 3 | ^k o ¹

k ,

PR

3

⁼

{

⁽

t 1 , t 2 , t 3

⁾

∈ R ³

^:

F

⁽

t 1 , t 2 , t 3

⁾

≤

¹

}.

It is assumedthat

κ, k ∈ N

^, ²

≤ κ ≤ k

^,

k >

^3,

κ

^a^divisor^of

k

^. ^Then ^we ^have

abodyof revolutionfor

κ

⁼²^and^a^super^sphere^for

κ

⁼

k

^. ^Therefore,^we^call

PR

3

^a^body ^of^pseudo^revolutionⁱⁿ^general.

Weconsiderthepoints(

t 1 , t 2 , t 3

⁾ât^the^boundaryând ^weâre^conned^to ^the

points

t 1 , t 2 , t 3 ≥

⁰^without^loss^ofgenerality. Weput

F

⁽

t 1 , t 2 , t 3

⁾⁼¹

, t 3

⁼

f

⁽

t 1 , t 2

⁾

,

where

f

^is^given^by

f

⁽

t 1 , t 2

⁾⁼

1

−

⁽

t ^κ 1

⁺

t ^κ 2

⁾

k κ

¹

k .

The

Gaussian

curvaturein suchapointisdenedby

K

⁼

H

⁽

f

⁽

t 1 , t 2

⁾⁾

(1+

f _t ² ₁

⁽

t 1 , t 2

⁾⁺

f _t ² ₂

⁽

t 1 , t 2

⁾⁾

2

(2)

where

H

⁽

f

⁽

t 1 , t 2

⁾⁾^denotes^the

Hessian

H

⁽

f

⁽

t 1 , t 2

⁾⁾⁼

f t 1 t 1

⁽

t 1 , t 2

⁾

f t 2 t 2

⁽

t 1 , t 2

⁾

− f _t ² ₁ _t ₂

⁽

t 1 , t 2

⁾

.

Inthepresentcasewendbymeansof longbutsimplecalculations

K

⁼ ⁽

k −

¹⁾⁽

κ −

¹⁾⁽

t 1 t 2

⁾

κ− 2

(

t ^κ 1

⁺

t ^κ 2

⁾

2 ^k _κ − 2

t ^k− 3 ²

(

t ² ₃ ^k− ²

⁺⁽

t ² ₁ ^κ− ²

⁺

t ² ₂ ^κ− ²

⁾

² ^k ^κ ⁻ ²

⁾

² .

From thisit isseenthat wehavethefollowingpointswith

Gaussian

curvature

K

⁼⁰^at^the^boundary:

(1)Bodyofrevolution(

κ

⁼^2): ^The^curve

t ² 1

⁺

t ² 2

⁼¹

, t 3

⁼⁰

andtheisolatedpoints(

t 1 , t 2 , t 3

⁾⁼⁽⁰

,

⁰

, ±

^1).

(2)Supersphere(

κ

⁼

k

^): ^The^curves

t ^k 2

⁺

t ^k 3

⁼¹

, t 1

⁼^0;

t ^k 1

⁺

t ^k 3

⁼¹

, t 2

⁼^0;

t ^k 1

⁺

t ^k 2

⁼¹

, t 3

⁼⁰

.

(3)Bodyofpseudorevolution(2

< κ < k

^): ^The^curves

t ^k 2

⁺

t ^k 3

⁼¹

, t 1

⁼^0;

t ^k 1

⁺

t ^k 3

⁼¹

, t 2

⁼^0;

t ^κ 1

⁺

t ^κ 2

⁼¹

, t 3

⁼⁰

.

Theatpoints(

t 1 , t 2 , t 3

⁾⁼⁽⁰

,

⁰

, ±

¹⁾^are^ofexceptionalimportanceinallthree casesandthepoints(

t 1 , t 2 , t 3

⁾⁼⁽

±

¹

,

⁰

,

^0),⁽

t 1 , t 2 , t 3

⁾⁼⁽⁰

, ±

¹

,

⁰⁾^are^meaningful

aswellinthecases(2)and (3).

Theaimofthepaperistoinvestigatethenumberoflatticepointsinthedilated

bodyofpseudorevolution

x

^PR

3

^,^that^is:

(1)

A _k,κ

⁽

x

^;^PR

3

⁾⁼^#

n

(

n 1 , n 2 , n 3

⁾

∈ Z ³

^:⁽

|n 1 | ^κ

⁺

|n 2 | ^κ

⁾

^k ^κ

⁺

|n 3 | ^k ≤ x o

.

Especiallywestudytheinuence ofthepointswith

Gaussian

curvaturezeroto theasymptoticrepresentationof

A _k,κ

⁽

x

^;^PR

3

^).

In[3]adetaileddescriptionisgivenforthecaseofsuperspheres. Seealsothe

paper[5]. Therefore,weareherein therstplaceinterestedforthecase

κ < k

^,

butwedonotexcludethecase

κ

⁼

k

^.

It is not too hard to obtain the following asymptotic representation for

A _k

⁽

x

^;^PR

3

⁾^from^the^results^of^the^paper^[7]:

A _k,κ

⁽

x

^;^PR

3

⁾⁼^vol(PR

3

⁾

x ³

⁺

H _k,κ, 1

⁽

x

⁾⁺

H _k,κ, 2

⁽

x

⁾⁺

O

x ⁵ ³ ⁻ ^3k ²

+

O

x ³ ²

^log

³ x

.

(3)

Thesecond main term

H _k,κ, 1

⁽

x

⁾îs â^certain^function ôf

x

^coming^from ^the ^at

points(

t 1 , t 2 , t 3

⁾⁼⁽⁰

,

⁰

, ±

¹⁾^and^can ^be^estimated^by

H _k,κ, 1

⁽

x

⁾

x ² ⁻ ² ^k .

Analogously,thethirdmainterm

H _k,κ, 2

⁽

x

⁾îsâ^certain^functionôf

x

^coming^from

theatpoints(

t 1 , t 2 , t 3

⁾⁼⁽

±

¹

,

⁰

,

⁰⁾

,

⁽⁰

, ±

¹

,

⁰⁾^and^can^be^estimated^by

H _k,κ, 2

⁽

x

⁾

x ² ⁻ ¹ ^κ ⁻ ¹ ^k .

Thersterrortermresultsfromtheotherpointswith

Gaussian

curvaturezero andtheseconderrortermresultsfromthepointswith

Gaussian

curvaturenon- zero.

Inthispaperwewillgiveexplicitrepresentationsofthesecondandthirdmain

terms which automatically show that the above upper bounds are at the same

time lower bounds. Further we give an improved estimation of the rst error

term.

Letthegeneralized

Bessel

functions

J _ν ⁽ ^k ⁾

⁽

x

⁾^be^dened ^by

(2)

J _ν ⁽ ^k ⁾

⁽

x

⁾⁼

√

²

π

⁽

ν

⁺¹

− ¹ _k

⁾

x

2

^kν

2 Z ₁

0

1

− t ^k _ν− ¹

k

cos

xt dt,

where isthegammafunction,

k, ν

^are^real^numbers^with

k ≥

^1,

ν > ¹ _k

^. ^F^urther

let

(3)

ψ ⁽ ν ^k ⁾

⁽

x

⁾⁼²

√ π

ν

⁺¹

−

¹

k

X _∞

n =1

x πn

^kν

2 J ν ⁽ ^k ⁾

⁽²

πnx

⁾

,

whichisabsolutelyconvergentfor

ν > ¹ _k

^. ^For^a^proof^see ^[3].

Theorem1. Let

κ, k ∈ N

^,²

≤ κ ≤ k

^,

k >

^3,

κ

^a^divisor^of

k

^. ^Then

(4)

A _k,κ

⁽

x

^;^PR

3

⁾⁼^vol(PR

3

⁾

x ³

⁺

H _k,κ, 1

⁽

x

⁾⁺

H _k,κ, 2

⁽

x

⁾⁺

_k,κ

⁽

x

⁾

with

H _k,κ, 1

⁽

x

⁾⁼

2

(

1 κ

⁾

κ

⁽

_κ ²

⁾

ψ ⁽ ₃ ^k _/k ⁾

⁽

x

⁾⁼

O,

x ² ⁻ ^k ²

,

(5)

H _k,κ, 2

⁽

x

⁾⁼⁸

x Z ₁

0 t ^k

1

− t ^k ¹

k − 1

ψ ₂ ⁽ ^κ _/κ ⁾

⁽

xt

⁾

dt

⁼

O,

x ² ⁻ ^κ ¹ ⁻ ¹ ^k

,

(6)

k,κ

⁽

x

⁾

x ¹¹⁹ ⁷³ ⁻ ^146k ¹⁶⁵

^(log

x

⁾

³¹⁵ ¹⁴⁶

⁺

x ³ ²

^log

³ x.

(7)

(4)

2. Preparation of the problem

Wend,bysymmetry,

A _k,κ

⁽

x

^;^PR

3

⁾⁼¹⁶⁽

S 1 , 2 , 3

⁺

S 1 , 3 , 2

⁺

S 3 , 1 , 2

⁾⁺

O

⁽

x

⁾

,

where

S _i,j,k

^are^triple^sums

S _i,j,k

⁼

X

n 1

X

n 2

X

n 3

1

withthesummationconditions

SC

⁽

S _i,j,k

⁾^:⁰

≤ n i ≤ n j ≤ n _k ,

⁽

n ^κ 1

⁺

n ^κ 2

⁾

κ k

+

n ^k 3 ≤ x ^k , n _i

⁼⁰

, n _i

⁼

n _j , n _j

⁼

n _k

^get^a^factor ¹

2

.

Webeginthesummationprocessin eachsumwith

n _i

^.

Forexample,summingin

S 1 , 2 , 3

^over

n 1

^,^we^obtain

"

x ^k − n ^k 3

^κ

k − n ^κ 2

¹

κ #

+ 1

2 for

x ^k − n ^k 3

^κ

k − n ^κ 2

¹

κ < n 2

and

n 2

ôtherwise. Îf^weûse^[

y

^]⁺

¹ ₂

⁼

y −ψ

⁽

y

^),^we^get^a^term^which^can^be^written

asanintegral,andatermwiththe

ψ

^-function^as^a^remainder. ^Hence

16

S 1 , 2 , 3

⁼

S ₁ ⁽¹⁾ _, ₂ _, ₃

⁺

2 , 3

⁽

x

⁾⁺

O

⁽

x

⁾

.

Here

S ⁽¹⁾ ₁ _, ₂ _, ₃

⁼¹⁶

X

n 2

X

n 3

Z

t 1

dt 1

withthesummation-integrationconditions

SIC X

n 2

X

n 3

Z

t 1

!

:0

≤ t 1 ≤ n 2 ≤ n 3 ,

⁽

t ^κ 1

⁺

n ^κ 2

⁾

κ k

⁺

n ^k 3 ≤ x ^k ,

n 2

⁼

n 3

^gets^a^factor

1

2

and

(8)

2 , 3

⁽

x

⁾⁼

−

¹⁶

X

( n 1 ,n 2 ) ∈D 2,3

ψ x ^k − n ^k 3

^κ

k − n ^κ 2

¹

κ !

,

(5)

where

D 2 , 3

^denotes^the^domain

D 2 , 3

⁼

(

t 2 , t 3

⁾

∈ R ²

^:⁰

≤ x ^k − t ^k 3

^κ

k − t ^κ 2 < t ^κ 2 ≤ t ^κ 3

.

Inthenextstepwesumover

n 2

ⁱⁿ

S ₁ ⁽¹⁾ _, ₂ _, ₃

^and^obtain^similarly

16

S 1 , 2 , 3

⁼

S ₁ ⁽²⁾ _, ₂ _, ₃

⁺

P 2 , 3

⁽

x

⁾⁺

2 , 3

⁽

x

⁾⁺

O

⁽

x

⁾

with

S ₁ ⁽²⁾ _, ₂ _, ₃

⁼¹⁶

X

n 3

Z

t 1

Z

t 2

dt 1 dt 2 ,

SIC X

n 3

Z

t 1

Z

t 2

!

:0

≤ t 1 ≤ t 2 ≤ n 3 ,

⁽

t ^κ 1

⁺

t ^κ 2

⁾

k κ

⁺

n ^k 3 ≤ x ^k

and

P 2 , 3

⁽

x

⁾⁼

−

¹⁶

X

n 3

Z

t 1

ψ x ^k − n ^k 3

^κ

k − t ^κ 1

¹

κ ! dt 1

(9)

SIC X

n 3

Z

t 1

!

: 0

≤ t 1 ≤ n 3 ,

x ^k − n ^k 3

^κ

k − n ^κ 3 < t ^κ 1 ≤

¹

2

x ^k − n ^k 3

^κ

k .

Inthelast stepwesumover

n 3

ⁱⁿ

S ₁ ⁽²⁾ _, ₂ _, ₃

^and ^nally^we^obtain

(10) 16

S 1 , 2 , 3

⁼

S ₁ ⁽³⁾ _, ₂ _, ₃

⁺

H 2 , 3

⁽

x

⁾⁺

P 2 , 3

⁽

x

⁾⁺

2 , 3

⁽

x

⁾⁺

O

⁽

x

⁾

,

where

S ₁ ⁽³⁾ _, ₂ _, ₃

⁼¹⁶

Z

t 1

Z

t 2

Z

t 3

dt 1 dt 2 dt 3 ,

(11)

IC Z

t 1

Z

t 2

Z

t 3

:0

≤ t 1 ≤ t 2 ≤ t 3 ,

⁽

t ^κ 1

⁺

t ^κ 2

⁾

k κ

+

t ^k 3 ≤ x ^k ,

H 2 , 3

⁽

x

⁾⁼

−

¹⁶

Z

t 1

Z

t 2

ψ

x ^k −

⁽

t ^κ 1

⁺

t ^κ 2

⁾

k κ

¹

k

dt 1 dt 2 ,

(12)

IC Z

t 1

Z

t 2

=0

≤ t ^k 1 ≤ t ^k 2 ≤ x ^k −

⁽

t ^κ 1

⁺

t ^κ 2

⁾

k κ .

(6)

Inthesamewayweobtainfortheothertriplesums:

(13) 16

S 1 , 3 , 2

⁼

S ₁ ⁽³⁾ _, ₃ _, ₂

⁺

H 3 , 2

⁽

x

⁾⁺

P 3 , 2

⁽

x

⁾⁺

3 , 2

⁽

x

⁾⁺

O

⁽

x

⁾

,

where

S ⁽³⁾ ₁ _, ₃ _, ₂

⁼¹⁶

Z

t 1

Z

t 2

Z

t 3

dt 1 dt 2 dt 3 ,

(14)

IC Z

t 1

Z

t 2

Z

t 3

: 0

≤ t 1 ≤ t 3 ≤ t 2 ,

⁽

t ^κ 1

⁺

t ^κ 2

⁾

k κ

⁺

t ^k 3 ≤ x ^k ,

H 3 , 2

⁽

x

⁾⁼

−

¹⁶

Z

t 1

Z

t 3

ψ x ^k − t ^k 3

^κ

k − t ^κ 1

¹

κ !

dt 1 dt 3 ,

(15)

IC Z

t 1

Z

t 3

: 0

≤ t ^κ 1 ≤ t ^κ 3 ≤ x ^k − t ^k 3

^κ

k − t ^κ 1 ,

P 3 , 2

⁽

x

⁾⁼

−

¹⁶

X

n 2

Z

t 1

ψ

x ^k −

⁽

t ^κ 1

⁺

n ^κ 2

⁾

κ k

¹

k dt 1 ,

(16)

SIC X

n 2

Z

t 1

!

: 0

≤ t ^k 1 ≤ x ^k −

⁽

t ^κ 1

⁺

n ^κ 2

⁾

k κ < n ^k 2 ,

3 , 2

⁽

x

⁾⁼

−

¹⁶

X

( n 3 ,n 2 ) ∈D 3,2

ψ x ^k − n ^k 3

^κ

k − n ^κ 2

¹

κ ! ,

(17)

D 3 , 2

⁼

(

t 3 , t 2

⁾

∈ R ²

^:⁰

≤ x ^k − t ^k 3

^κ

k − t ^κ 2 < t ^κ 3 ≤ t ^κ 2

.

Finally,weobtainfor

S 3 , 1 , 2

(18) 16

S 3 , 1 , 2

⁼

S ₃ ⁽³⁾ _, ₁ _, ₂

⁺

H 1 , 2

⁽

x

⁾⁺

P 1 , 2

⁽

x

⁾⁺

1 , 2

⁽

x

⁾⁺

O

⁽

x

⁾

,

where

S ⁽³⁾ ₃ _, ₁ _, ₂

⁼¹⁶

Z

t 1

Z

t 2

Z

t 3

dt 1 dt 2 dt 3 ,

(19)

IC Z

t 1

Z

t 2

Z

t 3

: 0

≤ t 3 ≤ t 1 ≤ t 2 ,

⁽

t ^κ 1

⁺

t ^κ 2

⁾

k κ

⁺

t ^k 3 ≤ x ^k ,

(7)

H 1 , 2

⁽

x

⁾⁼

−

¹⁶

Z

t 3

Z

t 1

ψ x ^k − t ^k 3

^κ

k − t ^κ 1

¹

κ !

dt 1 dt 3 ,

(20)

IC Z

t 3

Z

t 1

: 0

≤ t ^κ 3 ≤ t ^κ 1 ≤

¹

2

x ^k − t ^k 3

^κ

k ,

P 1 , 2

⁽

x

⁾⁼

−

¹⁶

X

n 2

Z

t 3

ψ x ^k − t ^k 3

^κ

k − n ^κ 2

¹

κ ! dt 3 ,

(21)

SIC X

n 2

Z

t 3

!

: 1

2

x ^k − t ^k 3

^κ

k < n ^κ 2 ≤ x ^k − t ^k 3

^κ

k − t ^κ 3 ,

1 , 2

⁽

x

⁾⁼

−

¹⁶

X

( n 1 ,n 2 ) ∈D 1,2

ψ

x ^k −

⁽

n ^κ 1

⁺

n ^κ 2

⁾

k κ

¹

k ,

(22)

D 1 , 2

⁼

n

(

t 1 , t 2

⁾

∈ R ²

^:⁰

≤ x ^k −

⁽

t ^κ 1

⁺

t ^κ 2

⁾

k κ < t ^k 1 ≤ t ^k 2

o .

3. Representationof the tripleintegrals

Itisclearthatitfollowsfrom (11),(14)and(19)

S ₁ ⁽³⁾ _, ₂ _, ₃

⁺

S ₁ ⁽³⁾ _, ₃ _, ₂

⁺

S ₃ ⁽³⁾ _, ₁ _, ₂

⁼^vol(PR

3

⁾

x ³ ,

whichisthemaintermin(4).

4. Representationand estimationofthe double integrals

Webeginwith

H 2 , 3

⁽

x

^). ^We^write^theintegrationconditionin(12)in theform 0

≤ t 1 ≤ t 2 , t ^κ 1

⁺

t ^κ 2 ≤

x ^k − t ^k 2

^κ

k .

Wehave

x ^k − t ^k 2

^κ

k ≥ x ^κ

²

⁻ ^κ ^k .

Weintegrateonlyupto

x ^κ ¹ ₂

^. ^The^remainderîsôfôrder

x

^. ^Hence,^bysubstituting

t 1 → t 1 x

^,

t 2 → t 2 x

^,^we^obtain

H 2 , 3

⁽

x

⁾⁼

−

¹⁶

x ²

Z

t 1

Z

t 2

ψ

x

1

−

⁽

t ^κ 1

⁺

t ^κ 2

⁾

κ k

¹

k

dt 1 dt 2

⁺

O

⁽

x

⁾

, IC

Z

t 1

Z

t 2

: 0

≤ t 1 ≤ t 2 , t ^κ 1

⁺

t ^κ 2 ≤

¹

2

.

(8)

Putting

t ^κ 1

⁺

t ^κ 2

⁼

z ^κ

^we^get,^by^symmetry^,

H 2 , 3

⁽

x

⁾⁼

−

⁸

x ²

Z

z ^κ ≤ ¹ ₂ z ^κ− ¹ ψ

x

1

− z ^k ¹

k dz

Z _z

0

(

z ^κ − t ^κ 1

⁾

κ 1 − 1

dt 1

⁺

O

⁽

x

⁾

=

−

⁸

²

⁽

¹ ^κ

⁾

x ² κ

⁽

_κ ²

⁾

Z

z ^κ ≤ ¹ ₂ zψ

x

1

− z ^k ¹

k

dz

⁺

O

⁽

x

⁾

=

−

⁸

²

⁽

¹ ^κ

⁾

x ² κ

⁽

_κ ²

⁾

Z ₁

0 z ^k− ¹

1

− z ^k ²

k − 1

ψ

⁽

xz

⁾

dz

⁺

O

⁽

x

⁾

,

whereagaintheintegralfrom 0upto(1

− z ^−k/κ

⁾

¹ ^/k

îsôfôrder

¹ _x

^. ^By^means^of

the

Fourier

representationofthe

ψ

^-function,

(23)

ψ

⁽

t

⁾⁼

−

¹

π X ∞ n =1

1

n

^sin(2

πnt

⁾

dt,

weobtain

H 2 , 3

⁽

x

⁾⁼

8

2

(

1 κ

⁾

x ² πκ

⁽

_κ ²

⁾

X ∞ n =1

1

n Z ₁

0 z ^k− ¹

1

− z ^k ²

k − 1

sin(2

πnxz

⁾

dz

⁺

O

⁽

x

⁾

= 8

2

(

1 κ

⁾

x ³ κ

⁽

² _κ

⁾

X ∞ n =1

Z ₁

0

1

− z ^k ²

k

cos(2

πnxz

⁾

dz

⁺

O

⁽

x

⁾

and,by(2)and(3),

H 2 , 3

⁽

x

⁾⁼⁴

√ π ²

⁽

_κ ¹

⁾ ⁽

_κ ²

⁺¹⁾

x ³ κ

⁽

_κ ²

⁾

X ∞ n =1

(

πnx

⁾

⁻ ³ ² J ₃ ⁽ ^k _/k ⁾

⁽²

πnx

⁾⁺

O

⁽

x

⁾

= 2

2

(

1 κ

⁾

κ

⁽

² _κ

⁾

ψ ₃ ⁽ ^k _/k ⁾

⁽

x

⁾⁺

O

⁽

x

⁾

.

Hence

H 2 , 3

⁽

x

⁾⁼

H _k,κ, 1

⁽

x

⁾⁺

O

⁽

x

⁾

and the representation (5) is obtained. The asymptotic representation of the

generalized

Bessel

functionsisgivenin Lemma3.11of[3]. Weobtain

J ₃ ⁽ ^k _/k ⁾

⁽²

πnx

⁾⁼

√ nx

( k

2

πnx ²

k

cos

2

πnx − π

2

k

⁺¹

+

O

1

nx )

.

(9)

Hence,itfollowswithapositiveconstant

c H _k,κ, 1

⁽

x

⁾⁼

cx ² ⁻ ² ^k

X ∞ n =1

n ⁻ ¹ ⁻ ^k ²

^sin

2

πnx − π k

+

O

⁽

x

⁾

.

Thus,theestimationsin(5)areclear.

Inorderto obtaintheterm

H _k,κ, 2

⁽

x

⁾^we^add

H 3 , 2

⁽

X

⁾^and

H 1 , 2

⁽

x

^). ^Then^we

getfrom(15)and(20)

H 3 , 2

⁽

x

⁾⁺

H 1 , 2

⁽

x

⁾⁼

−

¹⁶

x ² Z

t 1

Z

t 3

ψ x

1

− t ^k 3

^κ

k − t ^κ 1

¹

κ !

dt 1 dt 3 ,

IC Z

t 1

Z

t 3

: 0

≤ t ^κ 1 ≤

¹

2

1

− t ^k 3

^κ

k ,

⁰

≤ t ^κ 3 ≤

1

− t ^k 3

^κ

k − t ^κ 1 .

Bymeansofthesubstitution

t ^κ ₁

⁼⁽¹

− t ^k ₃

⁾

^κ/k − z ^κ

^we^obtain

H 3 , 2

⁽

x

⁾⁺

H 1 , 2

⁽

x

⁾⁼¹⁶

x ²

Z

z

Z

t 3

z ^κ− ¹

1

− t ^k 3

^κ

k − z ^κ ¹

κ − 1

ψ

⁽

xz

⁾

dt 3 dz, IC

Z

z

Z

t 3

: 1

2

1

− t ^k 3

^κ

k ≤ z ^k , t ^k 3

⁺

z ^k ≤

¹

,

⁰

≤ t 3 ≤ z.

Wemayextendthedomainofintegrationsuchthatthenewintegrationconditions

aregivenby

IC Z

z

Z

t 3

:

t ^k 3

⁺

z ^k ≤

¹

,

⁰

≤ t ^k 3 ≤

¹

2

.

The integral overthe new domain is of order

1 x

^which ^can ^be ^seen ^by ^partial

integratingwithrespectto

z

^. ^Now^we^substitute¹

− t ^k 3

⁼

t ^k

^. ^Then

H 3 , 2

⁽

x

⁾⁺

H 1 , 2

⁽

x

⁾⁼

=

−

¹⁶

x ² Z ₁

2 ^−1/k dt Z _t

0 t ^k− ¹

1

− t ^k ¹

k − 1

z ^κ− ¹

⁽

t ^κ − z ^κ

⁾

^κ ¹ ⁻ ¹ ψ

⁽

xz

⁾

dz

⁺

O

⁽

x

⁾

.

Thesubstitution

z → zt

^gives

H 3 , 2

⁽

x

⁾⁺

H 1 , 2

⁽

x

⁾⁼

=

−

¹⁶

x ² Z ₁

2 ^−1/k t ^k

1

− t ^k ¹

k − 1

dt Z ₁

0 z ^κ− ¹

^{( 1}

− z ^κ

⁾

¹ ^κ ⁻ ¹ ψ

⁽

xtz

⁾

dz

⁺

O

⁽

x

⁾

.

(10)

Wetaketheintegralwithrespectto

t

^from ⁰^up^to ^1,^since^the^new^part^of ^the

integralisof order

1 x

^. ^Asⁱⁿ ^case^of

H 2 , 3

⁽

x

⁾ ^we^use^the

Fourier

representation (23)ofthe

ψ

^-function^and^we^obtainanalogously

H 3 , 2

⁽

x

⁾⁺

H 1 , 2

⁽

x

⁾⁼

=

√

16

π

1

k

⁺¹

x ² Z ₁

0 t ^k ⁺¹

1

− t ^k ¹

k − 1 X ∞ n =1

1

n J ₂ ⁽ ^κ _/κ ⁾

⁽²

πnxt

⁾

dt

⁺

O

⁽

x

⁾

=8

x Z ₁

0 t ^k

1

− t ^k ¹

k − 1

ψ ₂ ⁽ ^κ _/κ ⁾

⁽

xt

⁾

dt

⁺

O

⁽

x

⁾

.

Hence

H 3 , 2

⁽

x

⁾⁺

H 1 , 2

⁽

x

⁾⁼

H _k,κ, 2

⁽

x

⁾⁺

O

⁽

x

⁾

and the representation (6) is obtained. For the asymptotic representation we

use(24). Clearly,theintegralfrom0upto

1

2

îsôfôrder

1 x

^. ^Therefore,^we^use^the

asymptoticrepresentationof thegeneralized

Bessel

function from Lemma 3.11 in[3]for

t ≥ ¹ ₂

^. ^Then

J ₂ ⁽ ^κ _/κ ⁾

⁽²

πnxt

⁾⁼

√

¹

π

κ

2

πnxt ¹

κ

cos

2

πnxt − π

2

1

κ

⁺¹

+

O

1

nx

.

Hence,itfollowswithapositiveconstant

a H _k,κ, 2

⁽

x

⁾⁼

=

ax ² ⁻ ^κ ¹ X ∞ n =1

n ⁻ ¹ ⁻ ¹ ^κ Z ₁

1 2

t ^k ⁺¹ ⁻ ^κ ¹

1

− t ^k ¹

k − 1

sin

2

πnxt − π

2

κ

dt

⁺

O

⁽

x

⁾

.

The remaining integral has a singularity at

t

⁼ ^1. ^We ^obtain ^the ^asymptotic

representationoftheintegralveryeasybymeansofChapter3,Section11from[1].

We use a special case of formula (11.6) on page 24: Let

φ

⁽

t

⁾ ^be continuously dierentiablein

α ≤ t ≤ β

^. ^Let

φ

⁽

α

⁾⁼^0. ^Then,^if ⁰

< µ <

^1,

Z _β

α e ^ixt

⁽

β − t

⁾

^µ− ¹ φ

⁽

t

⁾

dt

⁼ ⁽

µ

⁾

x ^µ e ^ixβ− ¹ ² ^µπi φ

⁽

β

⁾⁺

O

1

x

.

Thecondition

φ

⁽

α

⁾⁼⁰îs^not^necessary^. În^case ôf

φ

⁽

α

⁾

6

⁼⁰^the^point

α

^yields

anerrortermoforder1

/x

^. ^Thus, ^with^a^constant

b 6

⁼^0,^we^get

H _k,κ, 2

⁽

x

⁾⁼

bx ² ⁻ ^κ ¹ ⁻ ^k ¹

X ∞ n =1

n ⁻ ¹ ⁻ ¹ ^κ ⁻ ¹ ^k

^sin

2

πnx − π

2

1

κ −

¹

k

+

O

⁽

x

⁾

.

Hence,theestimations(6)followimmediately.

(11)

5. Estimationsof sumsand integrals

Theaimofthischapteristoestimatethesumsandintegrals(9),(16)and(21).

Lemma1. Assumethatforall

z

^with

¹ ₂ ≤ z ≤

¹^and^for^all

τ

^with

^x ₂ ^k ≤ τ ^k ≤ x ^k X

xk 2 <n ^k ≤τ ^k

ψ

x ^k − n ^k ¹

k z

_k, 2 , 3

⁽

x

⁾

.

Then

(25)

P 2 , 3

⁽

x

⁾

x ¹ ⁻ ^2κ ¹

_k, 2 , 3

⁽

x

⁾

₁ ₋ ¹

κ .

Proof:

Itiseasilyseenthatin(9)thecondition

t 1 ≤ n 3

^is^superuous^and^that

x ^k − n ^k 3

^κ

k − n ^κ 3 >

⁰

⇐⇒ n ^k 3 > x ^k

2

.

Further,consider

−

¹⁶

X Z

ψ x ^k − n ^k 3

^κ

k − t ^κ 1

¹

κ ! dt 1 ,

SIC XZ

: 0

≤ t ^κ 1 ≤

x ^k − n ^k 3

^κ

k − n ^κ 3 , x ^k

2

k/κ

+1

< n ^k 3 ≤ x ^k

2

.

It is seenat once that this termis of order

x

^. ^Now^we^bring ⁽⁹⁾ ^and ^this^term

togetherandobtain

P 2 , 3

⁽

x

⁾⁼

−

¹⁶

X Z

ψ x ^k − n ^k ₃ ^κ

k − t ^κ ₁ ¹

κ ! dt 1 ,

SIC XZ

: 0

≤ t ^κ 1 ≤

¹

2

x ^k − n ^k 3

^κ

k , x ^k

2

k/κ

+1

< n ^k 3 ≤ x ^k

2

.

Nowweput

t 1

⁼⁽

x ^k − n ^k ₃

⁾

¹ ^/k t

^. ^Then

P 2 ,k

⁽

x

⁾⁼

=

−

¹⁶

Z ₂ −1/κ

0

X

xk

2 k/κ +1 <n ^k ₃ ≤ ^xk ₂

x ^k − n ^k 3

¹

k ψ

x ^k − n ^k 3

¹

k

( 1

− t ^κ

⁾

¹ ^κ

dt

⁺

O

⁽

x

⁾

=

−

¹⁶⁽

I 1

⁺

I 2

⁾⁺

O

⁽

x

⁾

.

(12)

Applyingpartialsummationweobtainbymeansoftheconditionofthelemma

I 1

⁼

Z _y

0

X

xk

2 k/κ +1 <n ^k ₃ ≤ ^xk ₂

x ^k − n ^k 3

¹

k ψ x ^k − n ^k 3

¹

k

( 1

− t ^κ

⁾

¹ ^κ

dt

xy

_k, 2 , 3

⁽

x

⁾

.

In

I 2

^, ^where ^the ^integral ^is ^taken ^from

y

^up ^to ²

⁻ ¹ ^/κ

^, ^we ^use ^the

Fourier

expansionofthe

ψ

^-function⁽²³⁾^and ^obtain

I 2

⁼

1

π

X

xk

2 k/κ +1 <n ^k ₃ ≤ ^xk ₂

x ^k − n ^k 3

¹

k X ∞ m =1

1

m

Z ₂ −1/κ

y t ¹ ^−κ

⁽¹

− t ^κ

⁾

¹ ⁻ ¹ ^κ

× t ^κ− ¹

⁽¹

− t ^κ

⁾

¹ ^κ ⁻ ¹

^sin

2

πm

x ^k − n ^k 3

¹

k

(1

− t ^κ

⁾

¹ ^κ

dt

= 1

2

π ² X ∞ m =1

1

m ²

X

xk

2 k/κ +1 <n ^k ₃ ≤ ^xk ₂

 



t ¹ ^−κ

⁽¹

− t ^κ

⁾

¹ ⁻ ^κ ¹

^cos

2

πm

x ^k − n ^k 3

₁ ₋ ¹

k

(1

− t ^κ

⁾

^κ ¹

2 ^−1/k y

−

− Z ₂ −1/κ

y

d dt

t ¹ ^−κ

⁽¹

− t ^κ

⁾

¹ ⁻ ¹ ^κ

cos

2

πm

x ^k − n ^k 3

₁ ₋ ¹

k

(1

− t ^κ

⁾

^κ ¹

dt )

.

We estimate the sum over

n 3

^with

van der Corput’s

simplest theorem (see Theorem2.1in [3]). Thenwegetexactlyin thesamewayasonpage183of[3]

I 2 x ¹ ² y ¹ ^−κ .

Nowweput

y

⁼

x ¹ ²

_k, 2 , 3

⁽

x

⁾

₋ ¹

κ .

Thentheestimationsof

I 1

^and

I 2

âreêqualând^weôbtain^(25).

Lemma2. Assumethatforall

z

^with¹

≤ z ≤

²^and^for^all

u

^with

₁₊ ^x ^k _z k < u ^k ≤

x ^k X

1+zk xk <n ^k ≤u ^k

ψ

x ^k − n ^k z ^k ¹

k

_k, 3 , 2

⁽

x

⁾

.

(13)

Then

(26)

P 3 , 2

⁽

x

⁾

x ¹ ⁻ ^2κ ¹

_k, 3 , 2

⁽

x

⁾

₁ ₋ ¹

κ .

Proof:

Theproofisquitethesameas theprooftoLemma1suchthat weomit

it.

Lemma3. Assumethat

X

tκ 2 <n ^κ ≤t ^κ

ψ

(

t ^κ − n ^κ

⁾

¹ ^κ

_κ, 1 , 2

⁽

x

⁾

.

Then

(27)

P 1 , 2

⁽

x

⁾

x ¹ ⁻ ^2k ¹

κ, 1 , 2

⁽

x

⁾

₁ ₋ ¹

k . Proof:

Consider

−

¹⁶

Z _x

0

X

( x ^k −t ^k ₃ ) ^κ/k −t ^κ ₃ <n ^κ ₂ ≤ ( x ^k −t ^k ₃ ) ^κ/k

ψ x ^k − t ^k 3

^κ

k − n ^κ 2

¹

κ ! dt 3 .

It is easily seen that this term is of order

x

^. ^Now ^we ^bring ⁽²¹⁾^and ^this ^term

togetherandobtain

P 1 , 2

⁽

x

⁾⁼

=

−

¹⁶

Z _x

0

X

1 2 ( x ^k −t ^k ₃ ) ^κ/k <n ^κ ₂ ≤ ( x ^k −t ^k ₃ ) ^κ/k

ψ x ^k − t ^k 3

^κ

k − n ^κ 2

¹

κ !

dt 3

⁺

O

⁽

x

⁾

.

Putting

x ^k − t ^k ₃

⁼

t ^k

^then

P 1 , 2

⁽

x

⁾⁼

−

¹⁶

Z _x

0 t ^k− ¹

x ^k − t ^k ¹

k − 1 X

tκ 2 <n ^κ ₂ ≤t ^κ

ψ

(

t ^κ − n ^κ 2

⁾

κ 1

dt

⁺

O

⁽

x

⁾

.

Here wehavethe samesituation asin Lemma 4.8 of[3]. Using this result (27)

followsimmediately.

Estimations of the error terms

k, 2 , 3

⁽

x

^),

_k, 3 , 2

⁽

x

^),

κ, 1 , 2

⁽

x

^):

G. Kuba

[8]

haspointedoutthat

M.N. Huxley’s

[2]methodisapplicabletotheabovesums.

Assumethat

a, b, c, d

^are^xed^positive^real^numbers. ^Then^he^proved

X

( ^X _2b ) ^1/k −d<n≤ ( ^X−ack _b ) ^1/k −d

ψ



 X − b

⁽

n

⁺

d

⁾

^k a

! ¹

k 



√ X

ab ⁴⁶

73k





^log

X ²

ab _k ¹ 



315 146

.

Ekkehard Kr¨ atzel