1

A distributional way to prove the Goldbach conjectures

leveraging the circle method

Klaus Braun Jan 8, 2015

www.riemann-hypothesis.de

Abstract

Hardy-Littlewood [HaG] developed an analytical method, called circle method, which concerns with additive prime number problems. The circle method is about Fourier

analysis over Z , which acts on the circle ZR / . The analyzed functions are complex-

valued power series

0

)( n

n zaxf , 1z .

The fundamental principle is ([ViI] chapter I, lemma 4, Notes)

1

0

int22 )( dterefar it

n

n , 10 r .

The circle method is applied to additive prime number problems. Hardy-Littlewood [HaG]

resp. Vinogradov [ViI] applied the Farey arcs resp. major and minor arcs ([HeH]) to derive estimates for corresponding Weyl sums ([WaA]) supporting attempts to prove the 2-primes resp. 3-primes Goldbach conjectures. All those attempts require estimates for purely trigonometric sums ([ViI]), as there is no information existing about the distribution of the primes, which jeopardizes all attempts to prove both conjectures. This paper gives a conceptual design proposal to leverage the circle method to prove both Goldbach conjectures: it is proposed to replace the discrete Fourier transformation

applied for power functions )(xf by continuous Hilbert- ( H ), Riesz- ( A ) resp. Calderon-

Zygmund-transformations ( S ) (which are Pseudo Differential Operators of order 0 , 1

and 1 ) with distributional, periodical Hilbert space domains )1,0(#

H . The analogue

fundamental principle is

)()(:))((sin

)(

2

1)( 1

1

0

2xfAxSfdy

yx

yfxnf nn

n

n

for nybnyayf nnn 2sin2cos:)( .

This enables a discrete wavelet analysis (e.g.[GoJ], [LoA]) of a distributional prime

number density function on )(: 21 RS , resp. of the distributional von Mangoldt function

([ViJ])

)()()()( 2/1 RHnxnxxn

.

The distributional fractional part and )cot( x functions ensure the link to the Zeta

function in the Hilbert space framework )1,0(#

H . As mother function we propose the

Kummer function (resp. its derivative)

)2,

2

3;

2

1(:)( 11 ixFxb

with its relationship to the )(xerf function and its zeros nnxn ,2/1 . The translation

and scaling of the daughter wavelets are to be defined properly by the parameters

n andnp , which is not in scope of the paper.

2

Baseline and Solution Approach

The content of this section is an extract from [BrK1] [ViI1], [WoD]. The Hardy-Littlewood circle method ([HaG]) has been applied in additive number theory e.g. for

partition problems, i.e. an analysis about the number of additive partitions of N in a given number of integer summands.

The (3-primes, 2-primes) Goldbach conjectures are concerned with 3,2 .

The method is based on the generating power series for the prime number sequence series P

Pp

pzzF )( , 1z , Cz .

Depending from the number 3,2 of summands for 10 r it´s about an analysis of

derer

z

dzzzF

iNR iN

Pp

iN

rz

N 2

1

0

2)(2

1)(

fulfilling according to the Cauchy integral theorem

NN

NzNRzF )()( .

The key principle of the circle method is the fact, that for N being an integer it holds

otherwise

Nifde iN

0

0

11

0

2

which can be reformulated in the form

Lemma: Let

0

)( n

n xaxf with 1x , then for 10 r it holds

1

0

int22 )( dterefar it

n

n .

The corresponding properties for our approach ([BrK1]) to the above lemma are captured in

Lemma: Let nybnyayf nnn 2sin2cos:)( , then it holds for )1,0(x

)()(:))((sin

)(

2

1)( 1

1

0

2xfAxSfdy

yx

yfxnf nn

n

n

.

We note that

)(sin41 22

2 ie .

3

Lemma: Let

)2,2

3;

2

1(:)( 11 ixFxb

denote the Kummer function with its zeros nnxn ,2/1 and

)(: 21 RS . Then the transformation into an appropriate distribution

Hilbert space framework is enabled by the following three isometric mappings:

Re : , ixexe 2:

:j , )()(: nn pexej

RNk : , nxnk : .

The circle method is about Fourier analysis over Z , which acts on the circle ZR / . Fourier analysis

builds on periodical functions. It provides frequency-domains for corresponding analysis of the frequency of the original function. Fourier inverse then allows conclusions back to the origin function. The counterpart of FT to analyze non-periodical function is wavelet transforms, which is proposed to replace the arcs concept. The proposed corresponding Hilbert space framework is given by )1,0(#

H ( 0 ). The relationship to prime number theorem is given by ([ViJ])

ia

ia

s

xnxp s

dsx

s

s

inpx

n )(

)(

2

1)(log:)(

,

0

log

)()(

x

xdxJ

with

)()()()( 2/1 RHnxnxxn

and

)()(

)()()()( 2/1 RH

nx

nnxHnxH

xnxn

.

The relationship to the prime number distribution is given by the fractional function ([TiE]) resp. its Hilbert transform

1

2sin2

)(2:)( nxn

xxx

1

2cos2

)sin(2log2:)( nxn

xxH .

fulfilling the following properties

Lemma: In a distributional sense with appropriate domains )1,0()( #

HAD ,

)1,0()( #

1 HSD it holds

)1,0()(2cos2cos2

)()( #

2

1

LxiAnxinxn

xix H

n

H

)1,0()(2cos2cos2

#

2/1

1

HxiSnxinxn

i H

n

H

4

i.e.

),(),( iAi HH , )1,0(#

2L

2/12/1 ),(),( iSi HH , )1,0(#

2/1 H

. Lemma: For 1)Re(0 s corresponding Mellin transforms are given by

)()()(0

sxxdx

dxM

x

dxxxxss s

)(cos)()()2

cos()()( sMssMsss H

in a weak )1,0(#

2L sense

)2

sin()()(sin sssM

, )2

cos()()(cos sssM

.

From [LaE] we recall:

Let nG denote the number of all partitions of the even integer n in a sum of two primes p and q

(whereby qp and pq are counted as two different partitions) and )(xH the number of prime

pairs, for which xqp , then it holds

2

2

2

2log)log(log

)()()(

x

xp

x

u

du

ux

ux

u

duuxpxxH .

Every n can be represented as sum 21 nn in 1n different way. According to the prime number

theorem the probability that a selected integer is prime is about nlog/1 . Therefore an even n

seems to have nn log/ sums of two primes. In view of estimates of

1

0

/2)(m

mxiexS

we recall from [ViI] chapter 1, Lemma: Let )()()( xiQxPxF be a periodic function of x with period 1, and suppose that the

interval 10 x can be split up into finite number of intervals, such that the real functions )(xP

and )(xQ are continuous and monotonic in the interior of each. Suppose further that

)0()0(2

1)( xFxFxF

at each point of discontinuity of the function. Then

1

0 2sin2cos2

1)( nxbnxaaxF nn

with

1

0

2cos)(2 dnFan and

1

0

2sin)(2 dnFbn .

5

Remark: A sufficient condition that the Fourier series converge is the Dirichlet condition: The interval 1,0 is the union of finite intervals, where the function )(xF is continuous

For all points sx where )(xF is non continuous, )0( sxF and )0( sxF exist, then it holds

continuousxF

else

if

xFxF

xF

nxbnxaa nn

)(

)0()0(2

1

)(

2sin2cos2

1

1

0 .

Remark: The analysis of the fractional part function and its finite partial sums leads to the Gibbs phenomenon ([GrT]): the finite part sums have maxima and minima in the neighborhood of

sx , but the Fourier series

above is divergent at sx . Those sums are examples of continuous functions, where its

corresponding Fourier series is divergent at certain points. The situation is different in case of the

lower regularity assumption )1,0(#

2/1HF .

From [EdH] 1.11, 3.7, we recall Riemann’s prime number distribution function )(xJ definition

xpxp nn nnxJ

11

2

1:)(

and a special representation ( 1r ) of the von Mangoldt function with the zeros of the Zeta function in the form

1

2

0121

)(

n

xx

t

tdx

nx

.

It holds

00

)(

log

)(

log

)()(

xnr

xn

x

drn

nn

t

tdxJ

.

In a distribution framework the analogue of e.g. )(xJ is HxJS )(~

, which allows reduced

regularity assumptions to the domain, than the Dirichlet conditions. Remark: With respect to additive number theory problems and an alternative density definition we also refer to Schnirelmann density ([PeB]).

6

With respect to the Mellin and the Hilbert transforms of the Gaussian and Kummer function

2

:)( xexG , ),2

3;

2

1(:)( 2

11 xFxK

we recall from [BrK1]: Lemma: It holds

i.) )()(2)( ixKxxGxGH

ii.) dttGx

xK

x

0

)(1

)( resp. dxx

xKxGdK

)()(

iii.)

0

2/

1

)2/1(

s

sdKxs in the critical stripe

iv.)

)1(

)2

1()

2

1(

2

1)( 2/

s

ss

sGM s

H

.

Lemma: For the Hilbert transform HG of the Gaussian function G it holds

i.)

0

21

2

11)12(31

)2()1(2),

2

3;

2

1()(4)( n

nn

H xn

xxFxxGxG

ii.) The functions HG and G are identical in a weak ),(2 L .

Remark: The theorem of Erdös-Kac ([ErP]) concerning the Gaussian law of errors in the theory of additive number theoretic functions gave a first linkage from probability theory and additive number theory.

Remark: From ([BeB] 8, Entry 17 (iv), Entry (v)) we quote

“Ramanujan informs us to note that

)2sin(2)cot(:)(1

xxx ,

which also is devoid of meaning, .... may be formally established by differen-tiating the equality

)sin(2log2

2cos2

1

xx

. and that for

x

k k

kx

1

log:)(

it holds (“for the same limits”)

kkxxxx log)2sin(2)cot()2log(()()1(1

7

In the )1,0(#

H framework it holds (see also [ZyA] XIII, (11-3))

1

#

0

)2sin()(

2

1:))(( H

n

nxx

nxA n

i.e. )1,0(#

2/1H in a weak sense. The corresponding Hilbert and Calderon Zygmund Pseudo

Differential operators (of order 0 resp. 1) on the unit circle are given by

duAu )(

2sin2

1ln

2

1:)(

duHu )(

2cot

2

11:)(

d

uSu

2sin4

)(1:)(

2

.

The Hilbert space is linked to the Dirchlet series by the following ([LaE1] §227, Satz 40):

Theorem: The Dirichlet series

1

log:)( ns

neasf

1

log:)( ns

nebsg

are convergent for s ( 0 ). Then on the critical line it holds

1

2/1

1)2/1()2/1(

2

1lim:, nnba

ndtitgitfgf

.

The theorem is applied to derive Dirichlet series for )(

1

s and

)(

)2(

s

s

.

With respect related functions (e.g. von Mangoldt function) we use the following notations

)()( * xJxJ , )()( * xx , )()( * xLixLi .

As in the proposed framework the derivative operator is replaced by the Calderon-Zygmund operator with reduced domain regularity) this leads to the alternative density functions

)()( * xJSxJ , )()( * xSx .

Minor arcs are defined as the complementary set of the union of major arcs with respect to 1,0 . In

[HaG] only major (Farey) arcs are applied with center qa / and 2/1Nq . We note that

np n

e

p log2)

11(

.

The Fourier transformation provides (“just and only) frequency-domains in case of to be analyzed periodical function. To analyze non-periodical functions an appropriate transformation needs to have two properties: scaling and translation ([LoA]). This is given by the wavelet transform. Scaling

8

a mother wavelet is the process of compressing or expanding the wave. Translation is moving the wavelet along the inputted signal. The amount of translation that needs to be done depends on the scale of the wavelet. The smaller the scale the more translation that needs to occur. We note that e.g. the analogue Mexican hat mother wavelet has the form ([LoA] (1.1.3))

)

2(

2

1:)(

2

2

xx

dx

dx

.

We further note that

the Hilbert transform preserves orthogonality of translates and scaling relations ([WeJ]). The

Hilbert scale factor 2/1 of )1,0(#

2/1H is related to the wavelet theory ([GoJ], [LoA]) on the

unit disk (e.g. Daubechies and Möbius wavelets) the reproducing property (Calderón’s reproducing formula for the continuous wavelet transform) of Möbius wavelets is valid in a weak 2/1H sense.

The working assumption is that the arc analysis of the Weyl sums can be replaced by discrete wavelet analysis

)2(2:)( *

2/,

* kxx mmkm

in a )1,0(#

H framework with appropriately defined scaling and translation factors related to npn, .

The Kummer function ([AbM] 7.1.6)

)(ˆ1

:)12(31

)2()(

1)(

0

2

0

2

xgxn

xedttG

xxK

nnx

x

with

0)0( K and )()( xixgxG

is proposed to define an appropriate wavelet function with respect to ),(H framework.

We note the series representation of the Kummer functions

0

11!12

1),

2

3;

2

1(

n

x

nxF

n and 1

0

2/1

11

2/1

2 )2

1(

2

3

2

1

32),

2

1;0()(

n

n

xxFxxK

n

which are the solutions of the Kummer differential equation

02

1)

2

3(:)( uuxuxxuL .

With the notation from [NaS] it holds 0

2)( lxKx and therefore 2/1

2)( lxK and 2/1

2)( lxK .

For the expansion of the Kummer function in terms of Laguerre polynomials and Fourier transforms we refer to [PiA].

9

For an evaluation of the integral

0

)cos()(2

t

dxdtx

xtG

we refer to [GrI] 4.229, 8.232 with the formulas

1

2

0)!2(2

)1(log1cos

log)cos(

:)(kk

xxdt

t

txdx

x

xxci

kk

t

x

1

21

0)!2(2

)1(log)1

log(logkk

xxdt

t

kk .

We further note the identities

)()()( ixEixsixci

and )2

;2

3,

2

1()()( 2

11 xiFxziSzC

for the Fresnel integrals ([AbM] 7.3)

2

1)(

)14()!2(

)2/()1()

2cos(:)(

0

142

0

2

z

nnnz

zCznn

dttzC

2

1)(

)34()!12(

)2/()1()

2sin(:)(

0

3412

0

2

z

nnnz

zSznn

dttzS .

The conceptual idea is to transfer the Hardy-Littlewood method and its underlying framework into a distributional Hilbert space framework to enable approximation theory in the framework of functional analysis.

Definition: The function )(xe defines an isomorphism between R and the boundary of the unit

circle )(: 21 RS . We denote with x~ the transformed value of Rx and np~ being defined by

npi

nn epep~2

)~(

Let further nx be the zeros of

)2;2

3,

2

1(:)( 11 ixFxb .

Lemma: Putting

)1(:~ nxx nn

it holds

I) )()()(:)( xbAxibxbHxbH

II) 0)( nxb (“separation of zeros” theorem)

III) )2

1,0(~ nx .

We note that )(:)( nxexen builds an orthogonal system of )1,0(#

2L .

10

With respect to the zeros of the Zeta function we recall Lemma: Let )(TN the number of zeros of )(s in the region 10 , Tt 0 . Then for the

consecutive ordinates of the non-trivial zeros i of )(s it holds ([IvA] 1.4):

I) )(log22

log2

)( TOTTT

TN

II) n

nn

log

2

III) )1()1( nn NnN .

11

Summary The today´s Fourier analysis of Weyl (periodical, trigonometric) sums in a Banach space framework is proposed to be replaced by a discrete wavelet analysis on the circle in a distributional Hilbert space framework based on the (hypergeometric, non-periodical) Kummer function.

Related to an alternative distribution function of the primes

xlog )1,0()1,2

1()

2cot(

2log:)( #

2/1

#

2 HLxx

,

we propose the functions )(xb resp. )(xb as mother wavelet for a discrete wavelet analysis on

the circle in a distributional )1,0(#

2/1H Hilbert space, alternatively to the Hardy-Littlewood circle

method with )(xe and power series analysis on the unit disk 1: zZzD . The reason why

not be more generous than a Hilbert scale factor 2/1 is about the fact, that

2/1H .

We note that

),2/1(,

1

1

1

)))((

(1 nnx

else

nxx

px

nxxx

pxb nn

n

n

n

.

The discrete wavelet transforms (as weighted function with respect to the 2/1H inner product) is

proposed as alternative to the Farey resp. the major and minor arcs concept.

The properties of nx~ ,

np~ and nn xp ~~ are proposed to define appropriate translation and dilatation

factors for appropriate discrete wavelet transforms. The dilation factor triggers a contradiction of the analyzing (mother) wavelet function. The translation factor means a shift of the argument along the circle. We note that the index n counts the number of turns through the circle.

For the alternative prime number counting function xxx 1log)( we give a first trial

nx

nx

xx

xxx

nn

)2

tan(log

)2

tan(log)(

1

1

Remark: “S(chwartz)-asymptotics” of generalized functions are also called asymptotic by translation. Asymptotic by translation in combination with large scale dilations are also applied in [ViJ] to prove the prime number theorem in a distributional way.

12

The Goldbach Conjectures

A still missing proof of an additive prime number problem is the binary Goldbach conjecture, that

“all even numbers greater equal 4 can be expressed as the sum of two primes”. The (“weak”) ternary Goldbach conjecture (“the 3-primes problem) states, that

“every odd number greater equal 7 is a sum of three prime numbers”. Because of

3)1(212 nn

the ternary conjecture is a consequence of the binary conjecture. An equivalent formulation of the binary Goldbach conjecture is given by:

:NjNn jnp : and jnq : are prime numbers.

The conceptual idea of the below it about a transfer into appropriate distributional Hilbert space frameworks like

0

2

2/12/1 ),( AH , 0

2

11 ),( AAH

0

2

0 ),( HHHo

0

2

110 ),( SSH

being applied to singular integral equations if the form

d

xpxp

2

~sin4

)(~

~~1

2

.

13

A Distributional Way to the Prime Number Theorem ([ViJ])

“S(chwartz)-asymptotics” of generalized functions are also called asymptotic by translation. Asymptotic by translation in combination with large scale dilations are applied in [ViJ] to prove the prime number theorem in a distributional way: The Mangoldt function )(n defines the Chebyshev function ( 1a )

ia

ia

s

xnxp s

dsx

s

s

inpx

n )(

)(

2

1)(log:)(

,

0

log

)()(

x

xdxJ

with

)()()()( 2/1 RHnxnxxn

And )()(

)()()()( 2/1 RH

nx

nnxHnxH

xnxn

.

In order to prove the PNT xx )( the asymptotic behavior of )(x is analyzed by studying the

asymptotic properties of the distribution

)()log()(

:)( 2/1 RHnxn

nxv

xn

)()log(

)()( 2/1 RH

nxn

nxHv

xn

with its Fourier-Laplace transform for 0)Im( z

)1(

)1()(),(

11 iz

iz

n

netv

it

izt

.

This leads to the Fourier transform on the real axis, in the distributional sense,

p n

ixnppix

ixxv

1

)1( )(log)1(

)1()(ˆ

.

With respect to the asymptotic by translation and dilation we recall from [ViJ]

1)(lim

hxvh

in )(RS , 1)(lim

x

.

It further holds that

)()()(ˆ0

1)(ˆ 1 RLxYxv

ixxv loc

has tempered growth, whereby

ix

vPiyxix

xwy

1..

1lim:

0

1:)(

0

with )(2)(ˆ 2 xYexw x

14

being the disk boundary value of the analytic function z/1 , 0)Im( z ([PeB] I 316), and Y being the

Heaviside function with the integral representation

xi

x

ye

iy

d

idxY

2

1lim)()(

00

.

Recalling the definition of several Pseudo differential operators on the unit circles

duuA )(

2sin2

1ln

2

1:)(

~

duuH )(

2cot

2

11:)(

~

d

uuS

2sin4

)(1:)(

~

2

the analogue von Mangoldt and density distribution function of the primes are given by

)()~~()()(~2/1

Hnxnxxn

, )()~~()(~2/1 Hpxx

p

.

From the above it follows

xn

xSnx

nxH ))(~()~

)(1))(~(

, ))(~(~~

11))(~( xS

pxxH

p

.

The explicit formula of the latter equation is given by

d

xpxp

2

~sin4

)(~

~~1

2

.

15

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[PiA] Pichler A., Converging Series For The Riemann Zeta Function, arXix:1201.6538v1 [math.NT] 31 Jan 2012 [ScW] Schwarz W., Einführung in die Methoden und Ergebnisse der Primzahltheorie, BI, Hochschultaschenbücher, Mannheim, Wien, Zürich, 1969

[SeA] Sedletskii A. M., Asymptotics of the Zeros of Degenerate Hypergeometric Functions, Mathematical Notes, Vol. 82, No. 2 (2007) 229-237

[TiE] Titchmarsh E. C., The Theory of the Riemann Zeta-function, Clarendon Press, Oxford, 1951 [ViJ] Vindas J., Estrada R., A quick distributional way to the prime number theorem, Indag. Mathem., N.S. 20 (1) (2009) 159-165 [ViI] Vinogradov, I. M., Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk SSSR 15, 291-294 (1937) [ViI1] Vinogradov, I. M., The Method of Trigonometrical Sums in the Theory of Numbers, Dover Publications, Inc., Mineola, New York, 2004 [WaA] Walfisz A., Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der wissenschaften, Berlin, 1963 [WaG] Watson G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, Second Edition first published 1944, reprinted 1996, 2003, 2004, 2006 [WeJ] Weiss J., The Hilbert Transform of Wavelets are Wavelets, Applied Mathematics Group, 49 Grandview Road, Arlington, MA 02174 [WoD] Wolke D., Das Goldbach’sche Problem, Math. Semesterber. 41, 55-67 (1994) [ZyA] Zygmund A., Trigonometric series, Vol. I, Cambridge University Press, 1968

17

Appendix

Remark: An alternative dissection of the unit boundary circle is provided by the proof of a theorem in the context of conjugate function in the unit circle, which is given in [GaD] Chapter II, §1, in the context of the integral equation procedure of Theodorsen and Garrick for mapping circle-like boundaries on the unit circle the Hilbert transform:

Theorem: Let )()()( zivzuzf ( vu, real) be regular in the open unit disk

1z and continuous on the closed unit disk 1z . Then for )( iev the following

integral representation of )( ieu is valid (Cauchy integral):

deuvev ii )

2cot)(

2

1)0()(

2

0

.

The essential equation to prove this theorem are built on closed curves nC ([GaD]

Chapter II, §1), mainly on the unit circle, but with a arc within the disk according

to 0:

n

n

iez fulfilling

def

z

dz

ez

ezzf

ief i

C

i

ii

n

)2

cot)(2

1)(

2

1)(

2

0

.

For the polynomials nz on the unit circle this means

,..2,1

0

,...3,2,1

02

cot2

11

20

n

n

n

ie

ie

dein

in

in

.

18

The Hardy-Littlewood Circle Method

Hardy-Ramanujan and later Hardy-Littlewood [HaG] developed an analytical method, called circle method, which concerns with additive prime number problems. The basic idea of the circle method is an analysis of the generating power series with respect to the prime number series in the unit disk

Pp

pzzF )( , 1z , Cz

resp.

p

pzzzf )(log:)( .

Then e.g. the analysis of the binary and ternary Goldbach problem is about an analysis of

)(2 xF resp. )(3 xF . The knowledge of the distribution of the primes on the circle is essential.

For the ternary problem the analysis for the major arcs leads to the estimate ( 0A ,n odd)

)log

(2

)()())(log)(log(log:)(22

23

321

321n

nO

nnSdeSpppnR

A

in

npp

.

The proof is based on the (progressions prime number) theorem of Page, Siegel and Walfish ([ScW] VI, theorem 5.1):

)(log)(

1);( log

2

dc

xd

xeOt

dt

kkx

for x uniformly for all xk Alog

and an estimate for the “exponential sum”

np

pi

np

epeS 2:)(:)( .

Lemma: For real let 1;: Min . Then for integers NM , it holds ([ScW] VII,

lemma 2.2)

1)2(;)(

MNMinpeNnM

.

19

From [ViI1] X, lemma 1 and notes, we recall the Page theorem: Let ),;( lqN denote the number of primes p satisfying Np and )(mod qlp ,

where 1),( ql . Let 10 ,, cc be fixed positive numbers, and suppose that 10

crq .

Then, if q is not exceptional, we have

N

qr

NrO

x

dx

qlqN

2

))(

(log)(

1),;(

.

In case is near to a rational number qa / with 1crq for any fixed 1c , the Siegel theorem with

the estimate

N

NNeOx

dx

qlqN

2

)(log )(log)(

1),;(

2/1

enables a stronger approximation of

)( pe

without exceptional values of q .

The key idea of the circle method is to approximate 1r (e.g. by putting Ner /1 ) and to split the

interval 1,0 into “arcs” , i.e. intervals. Major arcs are intervals with a rational number as

its center qa / with “small” q : for qa / near a fraction qa / the number needs to be

small so that there are no other fractions with denominators q within a distance of of qa / .

This requirement leads to the Farey decomposition of the unit interval. The Farey fractions of order N are given by

1),(,0,1: qaqaNqq

aFQ

.

The Farey arcs around each of these fractions are defined as follows: let

q

a

q

a

q

a

be consecutive fractions in the Farey decomposition of order N. Then the intervals

qq

aa

qq

aaaqM Q ,:),( for

1

1

q

a , 0a

1

1,00,

1

11:)1,1(

QQM Q

are disjoint and their union covers the interval 1,0 .

20

In order to derive appropriate estimates for )(NR the knowledge of the distribution of the primes

is applied to analyze

p

prrF )( .

In order to prove the 3-prime Goldbach conjecture ( 3 ) for at least 43000

0 104.3 NN

([ChZ]) Vinograddov ([ViI]) replaced the infinite power series )(zF by the finite series (Weyl)

exponential sum ( R )

Np

ipeS 2:)( .

The major arcs were analyzed by using the Siegel-Walfisz prime number theorem

N

NNeOx

dx

qlqN

2

)(log )(log)(

1),;(

2/1

with

qp

qmqm p

qq )1

1(1:)(

1),(

.

For the remaining minor arcs related estimate the derived upper bound for )(S is purely built on

number theoretical arguments with respect to trigonometric sums ([ViI1]).

For the 2-prime conjecture Hardy-Littlewood [HaG1] gave the not proven relationship

))2

11(())

)1(

11((

log2)(

33222

Nppp ppN

NNR

and therefore

0)(2 NR for 0NN .

21

Let )(nN r ( )(NR ) denote the number of representations of n by a sum of primes and

npp

rr

r

ppn...

1

1

log...log:)( .

then I holds ([HaG] (1.45), (1.47))

2

)()( n

r

r znNzF resp.

2

)()( n

r

r znzf .

Applying the Cauchy integral formula it follows for 10 r , ([HaG] (5.21))

1

0

212 )(1)(

2

1)(

dereFrz

dz

z

zF

inR inir

n

rz

n

r

r .

The basic idea to prove appropriate approximation estimates was to approximate the boundary of

the unit circle by ner /1 applying the Farey dissection of the circle boundary with appropriately

defined major and minor arcs. Assuming that the zeros of all Dirichlet L-functions have real part less than ¾, Hardy-Littlewood applied the circle method to prove the 3-primes problem for sufficiently large odd numbers ([HaG], [LaE]). Vinogradov gave a proof w/o the Dirichlet L-functions assumptions, but “only” for sufficiently large n: Theorem (Vinogradov, 1937 [ViI]), every sufficiently large odd integer can be written as the sum of three primes (w/o a Dirichlet L-function assumption). Vinogradov proved an appropriate estimate of the major and minor arcs in the form

n

n

n

nR

npppprimppp

3

2

_,,3

3

log1

log

)(

321

321

,

For the major arcs of the binary problem Hardy-Littlewood ([HaG]) showed an appropriate estimate in the form

n

n

n

nnS

n

nnR

log

)(

log)(

log

2)(

2222

with

npnp pp

nS ))1(

11()

1

11(:)(

22 .

An analogue estimate of the minor arcs in the same way as for the ternary problem leads to a not sufficient estimate in the form

2/1

2log

)( nn

nnR

c ,

while a behavior same as for the major arcs or better would be required.

22

With respect to P1 and the 2L norm there is an alternative (still not sufficient) estimate in the form

(*) 2

1

2

min

22

log)(

n

nndeS

a

n

M arcsor

in

basically due to the Bessel inequality

nm

m vev22

, .

The Hurwitz Zeta function

A function, which is in a sense a generalization of )(s is the Hurwitz Zeta function, defined by

([TiE] 2.17:

0 )(

1:),(

nsan

as for 1)Re( s , 10 a .

For 1a resp. 2/1a this reduces to

)(s , )()12( ss .

There are also other generalized Zeta function, e.g. Lerch or Epstein or Dedekind Zeta function, as well as Zeta functions associated with cusp forms ([IvA] 11.8). Related to the Hurwitz Zeta function is the Dirichlet series ([IvA] 1.8), defined by

1

1

))(1()(

:),(

s

pns

ppn

nsL

for 1)Re( s

where for a fixed 0q )(n is the arithmetical function known as a character modulo q (for

)(),(,1 ssLq , if 1),( qa then 0)(( a ). For 1q it holds

qp

s

p

s

p

s pppssL 111

1 )1()1()1()(),( .

Thus ),( 1sL has a first-order pole at 1s just like )(s and it behaves similar to )(s in many

other ways, while ),( sL for 1 is regular for 0)Re( s .

The Generalized Riemann Hypothesis (GRH) states that all non-trivial zeros of all Dirichlet L-functions have real part equal to ½.

23

Notation and Formulas

With respect the notation and reference to the Riemann Hypothesis we refer to [KBr1]:

Let )(*

2 LH with )( 21 RS , i.e. is the boundary of the unit sphere. Let )(su being a

2 periodic function and denotes the integral from 0 to 2 in the Cauchy-sense. Then for

)(: 2 LHu with )(: 21 RS and for real Fourier coefficients and norms are defined by

dxexuu xi

)(2

1:

222

:

uu .

Then the Fourier coefficients of the convolution operator

dyyuyxkdyyuyx

xAu )()(:)(2

sin2log:))((

are given by

uukAu2

1)( .

The operator A enables characterization of the Hilbert spaces 2/1H and 1H in the form

0

2

2/12/1 ),( AH , 0

2

11 ),( AAH .

With respect to the Dirac function we note that building on the Dirichlet kernel there is a formal representation of )(x in the distribution sense in the form

),(),()(nsg2

1)cos(

1

2

1

2

1)( 12/1

0

2

0

HHxdkkxdkeex ikx

n

inx .

For

1

)2sin(2)cot(: n

it holds (see also [ZyA] XIII, (11-3))

1

#

0

)2sin()(

2

1))(( H

n

nxx

nxA n

.

24

From literature (e.g. [GaD] pp.63, [GrI] 1.441) we recall

,..2,1

0

,...3,2,1

2

10

2

1

2sin2

1ln

2

1

20

n

n

n

en

en

dein

in

in

,..2,1

0

,...3,2,1

02

cot2

11

20

n

n

n

ie

ie

dein

in

in

,..2,1

0

,...3,2,1

0

2sin4

11

20 2

n

n

n

ne

ne

dein

in

in

.

Due to the corresponding property of the Hilbert transform the functions H, are identical in a

weak )1,0(#

2L sense, i.e. it holds

i) 2

0

2

0H

ii) ),(),( H )1,0(#

2L

iii) 2/12/1 ),(),( H )1,0(#

2L

because of

2/10002/1 ),(),(),(),(),( AA HHH.

Remark: The functions of the Hardy space )(H

of 2L functions on the unit disk circle with

an analytical continuation inside the unit disk D can be parametrized by a point of Dz by

1

1:)(

izze

e

where the functions )(ze define a linear, continuous mapping according to

dyyz

f

idie

ez

f

ief i

iz

)(

2

1)(

2

1),(

,

which is an isometry of the spaces )(H

and )(DH

.

25

Remark: The dual space of 22/1

*

2/1 LHH is isometric to the classical Hardy space H2 of

analytical functions in the unit disc with norm

drefref i

H

i2

)(2

1:)(

2

.

It holds

i) If f H2 , then there exists boundary values ),()(lim)( 21

Lrefef i

r

i with

.

22

)(L

i

Heff

ii) If 2/1)( Heuef ii

, then its Dirichlet extension into the disc is given by ( irez ):

.

)()()(11

zuzueruzF i

with

2

2/1

22

0fuF

.

Remark: The Voronoi summation formula is related to the Dirichlet divisor problem. The Euler function )(n is defined as product of all prime divisors of n ([LaE])

np

nmnm p

nn )1

1(1:)(

1),(

Remark: From [SeA] we note that all zeros nz of the Kummer function

)(2

1

!12

1),

2

3;

2

1(

4

1,

4

1

2/4/3

1

0

2/1

0

11 zMezdtten

z

nzF zzt

n

n

lie in the half-plane 2/1)Re( z and in the horizontal stripe nzn 2)Im()12( . It holds

0

11

2/

1

)2

(

),2

3;

2

1(

s

s

x

dxxFxs .

Remark: With respect to the Kummer function we recall from [LoA] 1.1, [SeA]:

1. The zeros nz of the Kummer function )2(11 ixF lie in the intervals nn ,2/1

2. If )(111 RLF continuous and differentiable and )(: 211 RLF , then is a wavelet.

26

Remark: For the expansion of Kummer functions in terms of Laguerre polynomials and Fourier transforms we refer to [PiA]. Putting

dtex

xFxK

x

t

0

2

11

21);

2

3,

2

1(:)( and dtexG

x

t

2

:)(

the link between the polynomial function nz and the Hermite polynomials )(zH nis given by ([CaD]):

)()2

()( xdGx

izxH n

n

.

Alternatively we propose a corresponding polynomial system defined by

)()2

()( xdKx

izxK n

n

.

Lemma ([TiE] 3.7): Let )(xf be positive non-decreasing, and, as x let

x

xu

duuf

1

)( ,

then xxf )( .

Examples: xxxg log)()( and x

udug0

log)( .

27

Additional formulas

For 10 r , 2,0 , irez the function

k

k

k

kk

k

rk

kir

k

k

k

zrGzG

111

)sin()cos(:)(:)(

.

fulfills the properties:

i) 0)0( G

ii) 2

)2

sin2log()1(1

ik

eG

k

ik

iii) ri

i

k

kik

e

ererG

1)(

1

1

iv) ))1(Re()cos(

)2

sin2log(1

Gk

k

k

v) ))1(Im()sin(

2 1

Gk

k

k

vi)

11

1log)(

k

k

k

z

zzG for Dz .

It further holds ([GaD] §3):

0

2 )(:)( DLzazg k

k

0

2

)( 12 k

ag

k

DL

DLzazPnk

k

kn (:)( 2

nkn

k

DLnk

aPg 0

1

2

)(2

.

28

Theorems from Polya, Müntz, Ikehara, Wiener, Ramanujan, Nyman, Theodorsen,

Frullani, Hardy

Theorems from G. Polyá

G. Polyá obtained the following general theorem about zeros of the Fourier transform of a

real function:

Theorem 1: Let ba0 and let )(xg be a strictly positive continuous function on ),( ba

and differentiable there, except possible at finitely many points. Suppose that

)(

)(

xg

xgx

at every point of ),( ba where )(xg is differentiable. Suppose further that the integral

0

)(:)(x

dxxgxsG s

is convergent for ** )Re( sa . Then all zeros of )(sG in this stripe satisfy )Re( .

Let

1

1:)(

dxx

dx

sgs

s .

Theorem 2 (G. Polya): If is a polynomial which has all its roots on the imaginary axis, or

if is an entire function which can be written in a suitable way as limit of such polynomials,

then

If (*)

0

1 )(u

duuFu s has all its zeros on the critical line, so does

0

1 )(log)(u

duuuFu s .

Modern version: An operator which takes an even function )(q and replaces it by

)1()1( qq has the property of moving the zeros of a function closer on the imaginary

axis, and so an eigenfunction of this operator should have its zeros on the imaginary axis.

Theorem 3 (G. Polya): If )(xm is a polynomial which has all its roots on the imaginary axis,

or if it is an entire function which can be written in a suitable way as a limit of such polynomials, then

if

0

)( dxxFx s has all its zeros on the critical axis, so does

dyyeFedxxxFx yyss )()()(log)(0

.

29

The Müntz Formula

Theorem (Müntz’ formula): For )(),( xx continuous and bounded in any finite interval

with )()( xox and )()( xox for x and 1, it holds

x

dxdtt

xnxx

x

dxxxs ss

0100

)(1

)()(

)(

for 1)Re(0 s .

Proof: because )(x is continuous and bounded in any finite interval with )()( xox it

holds

1 0

1 )(1

dxxxn

s

s

exists for 1 ,

i.e. the inversion leading to the left hand side of (4.3) is justified.

ii) )1())(()1()()()()(/1

/1

0001

OdtxtOxdtOxdttttxdtxtnxx

x

The first summand is justified, because )(x is continuous and bounded in any finite interval

the second summand is justified, because )()( xox , i.e. it holds

x

cOnx

)1()(1

with dttc

0

)(: .

Hence

1)()()(

111

1

010

s

c

x

dxnxx

x

dx

x

cnxx

x

dxnxx sss

for 0 except 1s . Also

11

2

s

cdxxc s for 1

and therefore the result for 1)Re(0 s

30

Ikehara’s Theorem

If the measure d is positive and the function )(sg fulfills

i) )(sg is properly defined for 0)Re( s

ii) )(lim sg exists for 1s and is written as )1(g

iii) )1(

)1()(

s

gsg has a continuous extension from the open halfplane 1)Re( s , (whereby it is

necessarily defined and analytical) to the closed halfplane 1)Re( s ,

then

If 1lim

1

1

dxx

dx

s

s for 1s then 1lim 0

A

o

A

dx

d for A

i.e. roughly speaking dxd in the sense above. The function

)(

)()1(:)(

s

sssg

gives the prime number theorem. The Siegel formula (see below) might give the link to the

Stieltjes density above:

....2/)1(1)()(1(:)( ssssg .

From [LGa] we recall the two versions of Ikehara theorem:

Lemma (Ikehara version 1): Let be a monotone nondecreasing function on ),0( and let

1

1 )()(

x

xdxsF s .

If the integral converges absolutely for 1)Re( s and there is a constant A such that

1)(

s

AsF

extents to a continuous function in 1)Re( s then

Axx )( .

31

Lemma (Ikehara version 2): Let the Dirichlets series

1

)(s

n

n

csF

be convergent for 1)Re( s . If there exists a constant A such that

1)(

s

AsF

admits a continuous extension to the line 1)Re( s , then

N

n NAc1

* as N .

Wiener’s Tauberian theorem

The closed linear hull of the translates of a function

)()( 1 RLxf

is the whole space )(1 RL if and only if its Fourier transform

R

itx dttfexf )(:)(ˆ

never vanishes. Note that the close linear hull in question contains all convolutions

R

dyygyxfxgf )()(:)( .

32

Ramanujan’s Master Theorem

Ramanujan’s Master Theorem: In the neighborhood of 0x for

kxk

kxF )(

!

)()(

0

the following representation holds true

)()()( 1

0

ssdxxxF s

.

Ramanujan motivated his formula with the following wordings ([1] B. C. Berndt, chapter 4,

Entry 8):

“Statement: If two functions of x be equal, then a general theorem can be formed by simply

writing )(n instead of nx in the original theorem

Solution: “Put 1x and multiply it by )0(f then change x to ....,,, 432 xxxx and multiply

,...!3

)0(,

!2

)0(,

!1

)0( fff respectively and add up all the results. Then instead of nx we have

)( nxf for positive as well as for negative values of n . Changing )( nxf to )(n we can get the

result.”

Example: 2

1arctanarctan

xx

Ramanujan’s building process:

)0(2

1arctan1arctan)0( ff

,

2!1

)0(1arctanarctan

!1

)0( f

xx

f

,

2!2

)0(1arctanarctan

!2

)0( f

xx

f

...

33

Replace zarctan by its Maclaurin series in z , where z is any integral power of x . Now add

all the equalities above. On the left side one obtains two double series. Invert the order of

summation in each double series to find that

)1(212

)()()1(

0

1212

fn

xfxf nnn

.

Replace )( nxf by )(n to conclude that

)0(212

)12()12()1(

0

n

nnn .

Of course, this formal procedure is fraught with numerous difficulties, but the theorem was

finally correctly proved by G.H. Hardy.

The link to differential form is given by the Pfaff form dyydxx xy )()( .

Let

)0,0(: 2 RU

and 0)0,(: 2 xxRV

In case of the (non-star formed) domainU

there is no “integral” for the differential, but this is

the case for the domainV . In this case the “integral” of is related to one of the “core”

functions used by Ramanujan )(arctan( xy , which is

0

0

0

arctan2

arctan2

arctan

),(

),(

)0,1( y

y

x

x

yx

yx

y

yxF

yx

It holds

0),( yxdF

,

))(())(( y

xx

yxy

.

Remark Putting

)12(

1:)(

kk

the Hardy/Littlewood resp. the Riesz equivalence criteria of the Riemann Hypothesis are

(HL) RH holds if and only if )()(!

)()( 4/1

0

xOxk

kxF k .

(R) RH holds if and only if )()2()!1(

)1( 4/1

1

1

xOx

kk

kk

.

34

Bagchi’s Nyman criterion formulation

Let H denote the weighted 2l space consisting of all sequences Nnaa n of complex

numbers such that

1

2

n

nn a with 2

2

2

1

n

c

n

cn .

Let ,......1,1,11: , Hnk

nk

,....3,2,1)(: for ,...3,2,1k

and k be the closed linear span of k . Then the Nyman criterion states

The Riemann Hypothesis is true k

.

Integral Equation from Theodorsen

For a complex valued function 2 periodic function )()()( ivuf its conjugated

function can be represented by ([DGa], 1.1, 1.2)

2,0,

0 2cot)(

2

1

2cot)()(

2

1lim)( df

idff

if

.

Let nn baa ,;0 be the Fourier coefficients of f . Then

nn ab ,;0 are the Fourier coefficients of its

conjugate and it holds

2

0

2

2

0

2

0

2 )(1

2)(

1df

adf

resp. 2

1

2

2

0

2 )(1

n

n

n badf

.

Conformal mapping problem: Let C be a star-shaped Jordan curve of the w plane with

respect to 0w , let )( its representation by polar coordinates. Let D denote the inner

region of C and )(zfw the conformal mapping function with domain 1z onto D , which

is normalized by 0)0( f and 0)0( f . Then the unknown “boundary function” )( is

the solution of the non-linear, singular integral equation from Theodorsen:

2,0

2cot))((log

2

1)( d

i

.

This results in the following representation of the conformal mapping function

2,0

))((log2

1exp*)( d

ze

ze

izzf

i

i

1z .

35

Theorem of Frullani

Lemma (Theorem of Frullani) Let )(xf be a continuous, integrable function over any

interval BxA0 . Then, for ab 0 ,

0

log)0()()()(b

aFf

x

dxbxfaxf

where )(lim)0( xff for 0x and )(lim)( xff for x .

We mention a generalization of this lemma, due to Hardy (Quart. J. Math. 33 (1902) p. 113-

144) in the form

0

)(log)()(x

dxxbxax pnm .

The Hardy Theorem

The Gauss-Weierstrass density function

2

:)(:)(1

xexfx

with 1:

gives the Jacobi’s relation ([HEd] 1.6ff.)

)(21:/)/1()(:)(:)( 22 xxxGnxfxGx

.

A modified integral operator representation ([HEd] 11.1) in the form

0

1 11)(

)1(

)(2

x

dx

xxGx

ss

s s

is used to prove the Hardy theorem ([HEd] 11.1), i.e. that there are infinitely many roots of

0)( s on the line 2/1)Re( s . If the integral operator would be self-adjoint all zeros have to

be on the critical line.

36

References

[BBa] B. Bagchi, On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann Hypothesis, Indian Statistical Institute, Bangalore Centre, (2005), www.isibang.ac.in

[CBe] B. C. Berndt, Ramanujan´s Notebooks, Part I, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985

[HEd] H.M. Edwards, Riemann`s Zeta Function, Dover Publications, Inc., Mineola, New York, 1974

[DGa] D. Gaier, Konstruktive Methoden der konformalen Abbildung, Springer-Verlag, Berlin,

1964

[LGa] L. Garding, Some points of analysis and their history, University Lecture Series, no. 11, American Mathematical society, Providence, R.I., 1991

[ILi] I. K. Lifanov, L. N. Poltavskii, G. M. Vainikko, Hypersingular Integral Equations and their

Applications, Chapman & Hall/CRC, Boca Rato, London, NewYork, Washington, D. C., 2004

[GPo] G. Polya: Collected Papers, Volume II: location of zeros, edited by R. P. Boas, The MIT Press1974

[GPo1] G. Polya, Ueber eine neue Weise bestimmte Integrale in der analytischen

Zahlentheorie zu gebrauchen, Nachr. Ges. Wiss. Göttingen, 34, 149-159, (1917)

[GPo2] G. Polya, Ueber die Nullstellen gewisser ganzer Funktionen, Math. Zeit. 2, 352-383

(1918), also Collected Papers, II, 166-197

[GPo3] G. Polya, Bemerkungen über die Integraldarstellung der Riemannschen Zeta-

Funktion, Acta Mathematica 48, 305-317 (1926)

37

38

Hilbert Scales

There are certain relations between the spaces 0H for different indices:

Lemma: Let . Then

xx

and the embedding HH is compact.

Lemma: Let . Then

xxx for

Hx

with

and

.

Lemma: Let . To any Hx and 0t there is a )(xyy t according to

i)

xtyx

ii)

xyx ,

xy

iii)

xty )(

.

Corollary: Let . To any Hx and 0t there is a )(xyy t according to

i)

xtyx for

ii)

xty )( for .

Remark: Our construction of the Hilbert scale is based on the operator A with the two

properties i) and ii). The domain )(AD of A equipped with the norm

1

222,

i

ii xAx

turned out to be the space2H which is densely and compactly embedded in

0HH . It can

be shown that on the contrary to any such pair of Hilbert spaces there is an operator A with the properties i) and ii) such that

2)( HAD

0)( HAR and Axx

2.

39

For 0t we introduce an additional inner product resp. norm by

1

2

)( ),)(,(),(i

ii

t

t yxeyx i

2

)(

2

)(),( tt

xxx .

Now the factor have exponential decay tie

instead of a polynomial decay in case of

i .

Obviously we have

xtcx

t),(

)( for

Hx

with ),( tc depending only from and 0t . Thus the normt )( is weaker than

any norm . On the other hand any negative norm, i.e.

x with 0 , is bounded by the

norm0 and the newly introduced normt )( . It holds:

Lemma 5: Let 0 be fixed. The norm of any 0Hx is bounded by

2

)(

/2

0

22

t

t xexx

with 0 being arbitrary.

Remark 2: This inequality is in a certain sense the counterpart of the logarithmic convexity of the norm , which can be reformulated in the form ( 0, , 1 )

2/22

xexx

applying Young’s inequality to

)()(222

xxx .

The counterpart of lemma 4 above is

Lemma 6: Let 0, t be fixed. To any 0Hx there is a )(xyy t according to

i) xyx

ii) xy 1

1

iii) xeyx t

t

/

)(

.