The Circle’s Method to investigate the Goldbach’s Conjecture and the Germain primes: Mathematical connections with the p-adic strings and the zeta strings.

(1)

The Circle’s Method to investigate the Goldbach’s Conjecture and the Germain primes: Mathematical connections with the p-adic strings and the zeta strings.

Michele Nardelli1,2e Rosario Turco

1Dipartimento di Scienze della Terra

Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy

2Dipartimento di Matematica ed Applicazioni “R. Caccioppoli”

Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy

Abstract

In this paper we have described in the Section 1 some equations and theorems concerning the Circle Method applied to the Goldbach’s Conjecture. In the Section 2, we have described some equations and theorems concerning the Circle Method to investigate Germain primes by the Major arcs. In the Section 3, we have described some equations concerning the equivalence between the Goldbach’s Conjecture and the Generalized Riemann Hypothesis. In the Section 4, we have described some equations concerning the p-adic strings and the zeta strings. In conclusion, in the Section 5, we have described some possible mathematical connections between the arguments discussed in the various sections.

1. On some equations and theorems concerning the Circle Method applied to the Goldbach’s Conjecture [1]

Generating Functions on the Circle Method

(2)

altrimenti n n 0 a =1 se a ( ) 0 n n n N f z a z ∞ = ∈  = ⋅  

∑

Often are interesting only the representations at k at a time of numbers ai, the sum of these returns another number n; for which the k-ple obtained, as part of the set of all integers N, are only a subset of the Cartesian product Nk (or of N x N if k=2).

For example, for k=2 if n is the number sum belonging to N (set of the integers), and a1, a2 the numbers belonging to N, then the representations in pairs of numbers ai are:

{

}

2( ) : ( ,1 2) : 1 2

r n = a a ∈ ×N N n= +a a (1) In general with the Taylor’s series or with the residue Theorem, we obtain that:

2 1 1 ( ) ( ) 2 n f z r n dz i z π + =

_∫

(2)

i.e. r2(n) correspond with an of the series.

Having to work with a Cartesian product, it is possible also define (for the Cauchy’s product):

2 n 0 0 ( ) n n c = h k n n h k n h k n f z c z a a z ∞ ∞ = ≥ ≤ + = =

∑

⋅

∑ ∑

⋅ (3)

with akah≠1 if h and k belonging to N, then cn correspond to r2(n); thence, also here we have that: 2 2 1 1 ( ) ( ) 2 n f z r n dz i z

π

+ =

_∫

(4)

where the integral on the right is along a circle γ(ρ), path counterclockwise, centered in the origin and unit radius.

Is logic that in the case of Goldbach’s Conjecture we replace at N the set of primes P and the terms ai are belonging to P.

In general we have to deal with an additive problem with k>2, such as: the Waring’s problem (any k>2 and power s); the Vinogradov’s Theorem (for k=3 and with ai belonging to the set of the prime numbers P); the problem of the twin number for n=2, if –P={-2,-3,-5,-7,-11,…} and we consider P-P, studying the pairs (p1,p2) with p1,p2∫P such that n=p1-p2.

Then in the general case for k>2 is interesting the resolution of the equation of the type:

1 2 ... k n=a +a + +a

{

1 2 1 2

}

( ) : ( , ,..., ) k: ... k k k r n = a a a ∈N n=a +a + +a

(3)

0

( )

s n k n

f

z

r n z

∞ =

=

∑

for which: 1

1 ( )

( )

2

s k n

f

z dz

r n

i

z

π

+

=

_∫

(4’)

If we choose as generating function of the series the

₁ 0 1 ( ) (1 ) n n f z z z ∞ − = = = −

∑

Thence, if the power of the f is s=1, the eq. (4’) can be rewritten as follow:

1 1 ( ) 2 (1 ) k k n dz r n i z z

π

+ = −

∫

(5)

The eq. (5) of the general case is interesting because the integrand function has a easy singularity on γ(1); so it can easily integrate. In fact, for ρ<1 we can develop the k-th power of a binomial:

(

)

0 1 2 0 1 (1 ) ( ) ( ) ( ) ... ( ) ( ) 0 1 2 1 k m m k m k k k k k z z z z z z m m z ∞ − = − − − − −           = − =  − +  − +  − + +  − =   − −        

∑

 

If we replace in the eq. (5), we obtain:

₁ 1 0 1 1 ( ) ( 1) 2 (1 ) 2 m m n k k n m k dz r n z dz m i z z i

π

∞ − − + = −   = ₋ =   − =  

∑

∫

For m = n the integral is 2πi, while 0 in the other cases; thence the result is:

1 ( 1) 1 n k n k n k − + −     = −   = ₋      (6) Thence we can rewrite the eq. (6) as follows:

₁ 1 0 1 1 ( ) ( 1) 2 (1 ) 2 m m n k k n m k dz r n z dz m i z z i

π

∞ − − + = −   = ₋ =   − =  

∑

∫

1 ( 1) 1 n k n k n k − + −     = −   = ₋      (6b )

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The Vinogradov’s simplification

Vinogradov, however, made the further observation that to r2(n) can contribute only the integers m≤n; so we can introduce a different function (as opposed to one that led to eq. (6)), more useful to: 1 0 1 ( ) : 1 N N m N m z f z z z + = − = = −

∑

true for z≠1

For n≤N, for the Cauchy’s Theorem, we can write:

1 ( ) 1 ( ) 2 k N k n f z dz r n i z

π

+ =

_∫

(7)

Now fN(z) is a finite sum and there are no convergence problems, so the integrand function in the equation (7) has no points of singularity. Thence, now we can permanently fix the curve on which it integrates. Indeed, we take as curve the complex exponential function 2

( ) : ix

e x =eπ (1) and making a change of variable z = e(α) the eq. (7) becomes:

1 2 1 2 1 2 2 0 ( ) ( ) (( ) ) ( ) p N p N V

α

e n

α α

d e p p n

α α

d r n ≤ ≤ − =

∑ ∑

+ − =

∫

(8)

The eq. (8) is the n-th Fourier coefficient of f_Nk ( ( ))e α . If now we put T( )α = f_N ( ( ))eα , we obtain:

0 1 ( ) sin( ( 1) ) 1 (( 1) ) ₂ ( ) ( ) ( ( )) ( ) 1 ( ) sin( ) 1 N N N m e N N e N Z T T f e e m e N Z α π α α _αε α α α α _α _πα αε =  ₊  ₋ ₊  ₌ = = = = ₋  + 

∑

The function above now can be studied and is an element of elemental analysis exploitable. Property of TN(αααα)

The function T has peaks on the integer values and decreases much on the non-integer values. It’s easy to see, also as numerical computation, (see [3]) that:

1 1 | ( ) | min( 1, ) min( 1, ) | sin( ) | || || N T α N N πα α ≤ + ≤ + (9)

1_{We note that this function is orthogonal in the range [0,1] indeed:}

0 1 1 exp( ) exp( ) 0 altrimenti se a b az −bz dx= = 

∫

for which is 0 ( ) k(exp( )) exp( ) r n =

∫

f z −z dz

(5)

where ||α|| is the distance between two numbers, T is periodic of period 1 and α<sin(πα) for

αε(0,1/2]. If we use the eq. (9) and δ = δ(N) is chosen not too small, then the contribution in the range [δ,1-δ] is negligible. For example, if δ>1/N then:

1 1 1 1 2 | ( ) ( ) | | ( ) | || || 1 k k k N N k d T e n d T d k δ δ δ δ δ δ α α α α α α δ α − − − − − ≤ ≤ ≤ −

∫

(10)

Taking into account of eq. (8) and eq. (10) for n = N, k=2 e 1

( ) o N δ− ₌ _{we obtain that:} 2 2 1 1 2 0 1 sin( ( 1) ) sin( ( 1) ) ( ) 2 2 sin( ) sin( ) N N r n d d i δ δ

π

α

_α

π

α

_α

π

πα

−  ₊   ₊  =   =      

∫

(10b)

practically the Goldbach’s Conjecture via the Circle Method.

The Goldbach’s Conjecture, without to consider the difference between the weak and the strong conjecture, says that “given an even number greater than 4 this is always the sum of two prime numbers”.

For the Goldbach’s Conjecture, therefore, we are interested to the representations:

{

}

2( ) : ( 1, 2) : 1 2

r n = p p ∈ ×P P n= p +p

where p1, p2 are prime numbers not necessarily distinct, belonging to the set of the prime numbers P and for the moment we do not consider n as even number, but any (for example we accept also 2+3=5 at this stage of investigation). Putting:

( ) _N( ) ( )

p N V

α

V

α

e p

α

≤

= =

∑

(11)

then the Goldbach’s problem, with the techniques of real and complex analysis, results for n ≤ N:

1 2 1 1 2 1 2 2 0 0 ( ) ( ) (( ) ) ( ) p N p N V

α

e n

α α

d e p p n

α α

d r n ≤ ≤ − =

_{∑ ∑}

+ − =

∫

(12)

In the following, instead of consider directly the eq. (12), we can consider a weighted version with weight different from 1 (instead of consider p1+p2, we consider log(p1+p2)=logp1*logp2):

1 2 2

( ) :

log

1

log

2 p p n

R n

p

+ =

=

∑

(13)

It’s clear that r2(n) is positive if R2(n) is also positive; then it is sufficient to study R2(n) for the Goldbach’s conjecture.

(6)

( ) _N( ) log ( )

p N

S

α

S

α

p e p

α

≤

= =

∑

(14)

Bearing in mind the Dirichlet’s Theorem on the arithmetic progressions, choosing q, a such that MCD(q,a)=1, we write that

mod ( ; , ) log p N p a q N q a p

θ

≤ ≡ =

_∑

(15)

Theorem of Siegel - Walfisz

Let C,A>0 with q and a relatively prime, then

mod log ( ) ( ) logC p N p a q N N p O q N

ϕ

≤ ≡ = +

∑

for q≤logAN

the previous constant C does not depend on N, a, q (but more depend on A: C(A)).

Thence, from the Theorem of Siegel – Walfisz (see [4]) we have that:

( ; , ) ( ) ( ) logC N N N q a O q N

θ

ϕ

= + (16)

where we have defined

( ; , ) ( ) ( exp( ( ) log )) logC N E N q a O O N C A N N = = − .

where ϕ is the Euler totient function and C must be chosen not very large. The theorem is effective when q is very small compared to N. At this point, similarly to (12) we can write that for n≤N: 1 2 0 ( ) ( ) S

α

e −n

α α

d

∫

(17)

As preliminary operation, we see some value of S (for the exponential we recall the transformation

exp(2 )

x→ πix thence for example to ½ it comes to exp(2πi*1/2)=-1): S(0) = θ(N,1,1)≈N

S(1/2) = - θ(N,1,1)+2 log2≈-N

(7)

S(1/4) = exp(1/4) θ(N,4,1) + exp(3/4) θ(N,4,3) + log 2 ≈ 0

Now we see S also for some rational value a/q, when 0 ≤a≤q and MCD(a,q)=1. In this case the eq. (14) becomes: 1 mod ( / ) log ( / ) q h p N p h q S a q p e p a q = ≤ ≡ =

_{∑ ∑}

⋅ = * 1 1 1 mod

( / ) log = ( / ) ( ; , ) ( / ) ( ; , ) (log log )

q q q h p N h h p h q e h a q p e h a q θ N q a e h a q θ N q a O q N = ≤ = = ≡ =

_∑

⋅

_∑

⋅ ⋅ =

_∑

⋅ ⋅ + (18)

the asterisk in the last summation denote the further condition that MCD(h,q)=1. From the eq. (18) taking into account the eq. (16), we obtain:

* * 1 1 ( / ) ( / ) ( / ) ( ; , ) (log log ) ( ) q q h h N S a q e h a q e h a q E N q a O q N q ϕ = = =

∑

⋅ +

∑

⋅ ⋅ + * 1 ( ) ( / ) ( ; , ) (log log ) ( ) q h q e h a q E N q a O q N q µ ϕ = = +

_∑

⋅ ⋅ + (19)

where µ is the Moebius’s function(2).

For the Moebius’s function, |S(α)| is large when α is a rational number, in a neighborhood of a/q, and from the previous examples we have seen also that S(a/q) decreases as 1/q.

Realizing how about S, we can now try to find an expression for R2(n) and usually is used the “partial sum on the arcs”.

Putting: a

q

α = +η, for |η| small, we obtain:

( ) ( ) ( / ) ( ) ( ; , , ) ( ) ( ; , , ) ( )m N ( ) q q S a q e m E N q a T E N q a q q µ µ η η η η η ϕ ≤ ϕ + =

∑

⋅ = +

From the eqs. (16) and (19) we have that:

( ; , , ) A( (1 | |) exp( ( ) log ))

E N q a

η

=O q +N

η

N −C A N

2_µ

(q)=0 se q è divisibile per il quadrato di qualche numero primo, è (-1)k se q=p1p2...pk dove i pi sono k numeri primi

(8)

If as in [3] we denote with q,a a ( , ),q a a '( , )) q a

q−ξ q+ξ

M ( ) := ( the Farey’s arc concerning the rational number a/q, with ξ( , )q a and ξ'( , )q a of order (qQ)-1, then we define the set or the union of the Major and Minor arcs as follow:

[

]

* 1 := ( , ) m:= (1,1),1 (1,1) \ q q P a q a ξ ξ ≤ = +

U U

M M M (20)

Also here the asterisk indicates the additional condition that the MCD(q,a)=1. For the range of the Minor arc instead of consider [0,1] we have passed to

[

ξ

(1,1),1+

ξ

(1,1)

]

, that is possible for the periodicity 1.

It’s clear that, starting again to the eq. (17), now R2(n) is the sum of two integrals, one on the Major arc and the other on the Minor arc (as we have said when we have considered the eq. (12)) and for n≤N is:

1 2 2 2 0 ( ) ( ) ( ) ( ) ( ) ( ) R n =

∫

S α e −nα αd =

∫ ∫

+ S α e −nα αd = M m

How we have defined the Major arcs in the eq. (20), we have that '( , ) * 2 2 2 1 ( , ) ( ) ( ) ( ( )) ( ) ( ) ( ) ( ) q a q m q P a q a a a R n S e n d S e n d R n R n q q ξ ξ η η η α α α ≤ = ₋ =

∑∑ ∫

+ − + +

_∫

− = M + m (21)

In the following with the symbol ≈ we denote as in [3] an asymptotic equality (to the infinity). The eq. (21) can be rewrite also as follows:

'( , ) 2 * 2 2 1 ( , ) ( ) ( ) ( ) ( ( )) ( ) q a q q P a q a q a R n T e n d q q ξ ξ µ _η _{η η} ϕ ≤ = ₋ ≈

∑∑ ∫

− + = M '( , ) 2 * 2 2 1 _{( , )} ( ) ( ) ( ) ( ) ( ) q a q q P a _{q a} q a e n T e n d q q ξ ξ µ _η _{η η} ϕ ≤ = − =

∑

−

_∫

− (22)

If we extend the integral that contains T throughout the range [0,1] 1 2 1 2 0 ( ) ( ) 1 1 m m n T η e nη ηd n n + = − =

_∑

= − ≈

∫

(23) Thence, we obtain: 2 * 2 1 ( ) ( ) ( ) ( ) q q P a q a R n n e n q q µ ϕ ≤ = ≈

_∑

− M (24)

(9)

where the inner sum is called the Ramanujan’s sum and we can show it with a Theorem that can be expressed as a function of µ and ϕ:

2 2 2 ( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) ( , ) ( , ) q P q P q q q q q q n R n n n q q q q n q q n q n

µ

_µ

ϕ

µ

ϕ

_ϕ

ϕ

_ϕ

≤ ≤ ≈

∑

=

∑

M

If we extend the sum to q≥1 and we consider another Theorem (see [3]), we obtain:

2 1 ( ) ( ) ( , ) ( ) (1 ( )) ( ) ₍ ₎ ( , ) n q p q q q n R n n n f p q q q n

µ

ϕ

_ϕ

≥ ≈

∑

=

∏

+ 2 (25)

The “productor” is on all the prime numbers; further we have that:

2 2 1 se p|n ( ) 1 ( ) ( , ) ( ) 1 ( ) ₍ ₎ altrimenti ( , ) ₍ ₁₎ n q p q q n f p q q q n _p

µ

ϕ

_ϕ

  ₋  = = ₋  ₋ 

If n is odd then 1+ f_n(2)=0 thence the eq. (23) states that there aren’t Goldbach’s pairs for n. Indeed R2(n)=0 if n-2 is not prime number, R2(n)=2log(n-2) if n-2 is a prime number. If, instead, n is even, we can obtain the following expression:

(

)

2 | | 1 1 ( ) (1 ) (1 ) 1 ₁ p n p n R n n p _p ≈ + − − ₋

∏

2

(

)

(

)

2 2 | 2 | 2 2 1 1 1 ( ) 2 ( ) (1 ) ( ) 1 ( 2) ₁ o 2 p n p p n p p p p p R n n C n p p p _> _p p > > − − ≈ ⋅ − = − − ₋ −

∏

2 (26)

where C0 is the constant of the twin primes. The eq. (26) is the asymptotic formula for R2(n) based on the Number Theory and provides a value greater than r2(n) of a quantity ( log n )2, for the weights logp1logp2.

2. On some equations concerning the Circle Method to investigate Germain Primes [2]

In this section we apply the Circle Method to investigate Germain primes. As current techniques are unable to adequately bound the Minor arc contributions, we concentrate on the Major arcs,

(10)

where we perform the calculations in great detail. The methods of this section immediately generalize to other standard problems, such as investigating twin primes or prime tuples.

We remember the Siegel-Walfisz Theorem, that will be useful in the follow. Let C,B>0 and let a and q be relatively prime. Then

( )

∑

≡ ≤      + = q a p x p C x x O q x p log log

φ

. (27) Definition 1

A prime p is a Germain prime (or p and

2 1

−

p

are a Germain prime pair) if both p and

2 1

−

p

are prime. An alternate definition is to have p and 2p+1 both prime.

Let B,D be positive integers with D>2B. Set Q=logDN. Define the Major arc M_a,_q_{for each}

pair

( )

a,q with a and q relatively prime and B

q log 1≤ ≤ by       < −      ₋ ∈ = N Q q a x x q a : 2 1 , 2 1 , M (28) if 2 1 ≠ q a and      ₋     ₋ ₋ ₊ = 2 1 , 2 1 2 1 , 2 1 2 , 1 N Q N Q U M . (29)

We have that the our generating function is periodic with period 1, and we can work on either

[ ]

0,1 or _− _ 2 1 , 2 1

. As the Major arcs depend on N and D , we should write M_a_,_q

(

N,D

)

_and

(

N ,D

)

M . Note we are giving ourselves a little extra flexibility by having q≤logBN and each q a, M _{of size} N N D log

. By definition, the Minor arcs m are whatever is not in the Major arcs. Thus

the Major arcs are the subset of

    − 2 1 , 2 1

near rationals with small denominators, and the Minor arcs are what is left. Here near and small are relative to N . Then

(

)

∫

( ) ( )

∫

( ) ( )

∫

( ) ( )

− − = − + − = 2 1 2 1 2 1 , ; 1 M N m N N N N A F xe x dx F xe xdx F x e x dx A r . (30)

(11)

We chose the above definition for the Major arcs because our main tool for evaluating F_N

( )

x is the Siegel-Walfisz formula (see eq. (27)), which states that given any B,C>0, if q≤logBN and

( )

r,q =1 then

( )

∑

≡ ≤      + = q r p N p C N N O q N p log log

φ

. (31)

For C very large, the error term leads to small, manageable errors on the Major arcs.

Now we apply partial summation multiple times to show u is a good approximation to F on the _N

Major arcs M_a,_q_{. Define}

( )

( ) (

)

( )

2 2 q a c a c a C_q q q

φ

− = . (32) We show Theorem 1 For

α

∈M_a,_q_,

( )

_      +       − = ₋ N N O q a u a C F_N _q _C ₂_D 2 log

α

. (33)

The problem is to estimate the difference

( )

α

( )

α

( )

α

β

C

( ) ( )

a u

β

q a F q a u a C F S_a_q _N _q _N _− _q      + =       − − = , . (34)

To prove Theorem 1 we must show that

( )

N N S_a_q _C ₂_D 2 , log − ≤

α

. It is easier to apply partial summation if we use the λ-formulation of the generating function F because now both _N F and N

u will be sums over m₁,m₂ ≤N. Thus

( )

∑

( ) ( ) (

(

)

( )

∑

(

)

≤ ≤ − − − = N m m m m N q q a m m e m m C a e m m S 2 1, 1, 2 2 1 2 1 2 1 ,

α

λ

2

β

2

β

λ

( ) ( ) (

λ

)

( ) (

(

₁ ₂

)

β

)

, 2 1 2 1 2 2 2 1 m m e a C q a m m e m m N m m q  −      −       − =

_∑

≤

λ

( ) ( ) (

₁

λ

₂ ₁ ₂

)

( ) (

₂

β

) ( )

₁

β

1 2 2 2 C a e m em q a m m e m m N m m N q

∑ ∑

≤ ≤         −       −       − =

(12)

(

₁

) (

₁

) ( )

₁

β

1 2 2 2 m,N b m,N em a N m m N m m

∑ ∑

≤ ≤      =

∑

(

) ( )

≤ = N m q a m e m S 1 1 1 ,

α

;

β

, (35) where

(

) ( ) ( ) (

)

C

( )

a q a m m e m m N m am − q      − = 1 2 1 2 1, 2 2

λ

; bm2

(

m1,N

) (

=e−2m2

β

)

(

)

∑

(

) (

)

≤ = N m m m q a m a m N b m N S 2 2 2 , , ; ₁ ₁ ₁ ,

α

. (36)

Recall the integral version of partial summation states

∑

( ) ( ) ( )

∫

( ) ( )

= − = N m N mb m A N b N Au b u du a 1 1 ' , (37)

where b is a differentiable function and A

( )

u =

∑

_m_≤_uam. We apply this to am₂

(

m1,N

)

and

(

m N

)

b_m ₁, 2 . As

( ) (

)

2 2 4 2 2 2 m i m b m e m e

b = = −

β

= −πβ , b'

( )

m₂ =−4

π

i

β

e

(

−2

β

m₂

)

. Applying the integral version of partial summation to the m₂-sum gives

(

)

∑

( ) ( ) (

)

( ) (

)

∑

(

) (

)

≤ ≤ = = −       −       − = N m m N m m q q a C a e m a m N b m N q a m m e m m m S 2 2 2 2 , , 2 2 ; ₁ ₁ ₂ ₁ ₂ ₂ ₁ ₁ ,

α

λ

β

a

(

m N

) (

e N

)

i N a

(

m N

) (

e u

)

du u u m m N m m

β

π

β

 −

β

     + −       =

∑

_∫

∑

= _≤ ≤ 1 1 2 1 2 2 2 , 2 4 , . (38)

The first term is called the boundary term, the second the integral term. We substitute these into (35) and find

( )

(

) (

) ( )

+         −       =

_{∑ ∑}

≤ ≤

β

α

1 1 , 1 2 2 m,N e 2N e m a S N m m N m q a

∑

∫

∑

(

) (

)

( )

≤ = ≤         −       + N m N u u m m m N e u du e m a i 1 2 2 1 1 1, 4

π

β

. (39)

The proof of Theorem 1 is completed by showing S_a_,_q

( )

α

;B and S_a_,_q

( )

α

;I , where B = Boundary and I = Integral, are small. The first deal with the boundary term from the first partial summation on m2,Sa,q

( )

α

;B .

(13)

( )

_      = ₋ N N O B S_a_q _C _D log ; 2 ,

α

. (40)

Proof. Recall that

_,

( )

α

(

₁

) (

β

) ( ) (

₁

β

)

(

₁

) ( )

₁

β

1 2 2 1 2 2 , 2 2 , ;B a m N e N em e N a m N e m S N m m N m N m m N m q a

∑ ∑

≤ ≤ ≤ ≤       − =         −       = . (41)

As e

(

−2N

β

)

=1, we can ignore it in the bounds below. We again apply the integral version of partial summation with

∑

(

)

∑

( ) ( ) (

)

( )

≤ ≤      −       − = = N m m N q m m C a q a m m e m m N m a a 2 2 2 1 1,

λ

1

λ

2 1 2 2 ; bm1 =e

( )

m1

β

. (42) We find

(

) ( )

∑

(

) ( )

∫ ∑ ∑

(

) ( )

≤ ≤ = ≤ ≤      −       = N m N t t m m N m N m m q a B a m N e N i a m N et dt S N e 1 1 2 2 2 2 1 0 1 , ; , 2 , 2

β

α

β

π

β

. (43)

To prove Lemma 1, it suffices to bound the two terms in (43), which we do in Lemmas 2 and 3.

Lemma 2

(

) ( )

_      =      

∑ ∑

≤ ≤ N N O N e N m a _C N m m N m log , 2 1 1 2 2

β

. (44)

Proof. As e

( )

N

β

=1, this factor is harmless, and the m₁, m₂-sums are bounded by the Siegel-Walfisz Theorem.

(

)

( ) ( ) (

)

( )

_=      −       − =

∑

∑ ∑

≤N ≤ ≤ ≤ m m Nm N q N m m C a q a m m e m m N m a 1 2 1 2 2 1,

λ

1

λ

2 1 2 2

( )

−

( )

=            −             =

_∑

≤ ≤ 2 2 2 1 1 2 1 N a C q a m e m q a m e m _q N m N m

λ

( )

(

( )

)

−

( )

=           + − ⋅             + = 2 log 2 log N C a N N O q N a c N N O q N a c q C q C q

φ

_      = N N O _C log 2 (45) as

( )

( ) (

)

( )

2 2 q a c a c a Cq q q

φ

− = and c_q

( ) ( )

b ≤φ q .

(14)

Lemma 3

∫ ∑ ∑

(

) ( )

= _≤ _≤ − _     =       N t C D t m m N m N N O dt t e N m a i 0 2 1 log , 2 1 2 2 β β π . (46) Proof. Note N N N Q logD = ≤ β , and

( )

2

(

( )

2

)

2 q a c q a c a C_q q q

φ

−

= . For t≤ N , we trivially bound the 2

m -sum by N2 . Thus these t contribute at most

∫ ∑

= _≤ = ≤ N t t m D N N N Ndt 0 2 1 log 2 β β . (47)

An identical application of Siegel-Walfisz as in the proof of Lemma 2 yields for t≥ N ,

∑ ∑

(

)

≤ ≤ = t m m N m m N a 1 2 2 1,

( )

(

( )

)

−

( )

=            + − ⋅             + C a tN N N O q N a c N t O q t a c q C q C q log 2 log

φ

_      = N tN O _C log . (48) Therefore

∫

∑ ∑

(

)

= − ≤ ≤      =       = N N t C C D t m m N m N N O N N O dt N m a log log , 2 3 1 1 2 2 β β . (49)

We note also that:

(

) ( )

_⇒      =      

∫ ∑ ∑

₌ − ≤ ≤ N t C D t m m N m N N O dt t e N m a i 0 2 1 log , 2 1 2 2 β β π

∫

∑ ∑

(

)

= − ≤ ≤      =       = ⇒ N N t C C D t m m N m N N O N N O dt N m a log log , 2 3 1 1 2 2 β β . (50)

We now deal with the integral term from the first partial summation on m₂,S_a_,_q

( )

α;I . Lemma 4

( )

_      = ₋ N N O I S_a_q _C ₂_D 2 , log ;

α

. (51) Proof. Recall

( )

∑ ∫ ∑

(

) (

)

( )

≤ = ≤         −       = N m N u u m m q a I i a m N e u du e m S 1 2 2 1 1 1 , α; 4πβ , β β (52)

(15)

where

(

) ( ) ( ) (

)

C

( )

a q a m m e m m N m am − q      − = 1 2 1 2 1, 2 2 λ λ . (53)

Thence, the eq. (52) can be rewrite also as follow:

( )

∑ ∫ ∑

( ) ( ) (

)

( ) (

)

( )

≤ = ≤         −       −       − = N m N u u m q q a C a e u du e m q a m m e m m i I S 1 2 1 1 1 2 1 2 , α; 4πβ λ λ 2 β β . (53b)

We apply the integral version of partial summation, with

∫

∑

(

) (

)

= _≤ _ −     = N u u m m m a m N e u du a 1 1 2 2 1 , β bm1 =e

( )

m1β . (54) We find

( )

(

) (

)



( )

+      − =

∑ ∫ ∑

≤ = ≤ β β β π α I i a m N e u du e N S N m N u u m m q a 1 2 2 1 1 , ; 4 , N a

(

m N

) (

e u

)

du e

( )

mt dt t t m N u u m m 1 1 1 1 2 1 2 2 , 8

∫ ∑∫ ∑

= _≤ = _≤ _     − + πβ β . (55)

For the eq. (53), we can rewrite the eq. (55) also as follow:

( )

( ) ( ) (

)

( ) (

)



( )

+      − −       − =

∑ ∫ ∑

≤ = ≤ λ λ β β β π α C a e u du e N q a m m e m m i I S N m N u u m q q a 1 2 1 1 2 1 2 , ; 4 2

( ) ( ) (

)

C

( ) (

ae u

)

du e

( )

mt dt q a m m e m m N t t m N u u m q 1 1 1 1 2 1 2 2 1 2 2 8

∫ ∑∫ ∑

= _≤ = _≤ _     − −       − + πβ λ λ β . (55b)

The factor of 8πβ2 =−

(

4πiβ

) (

⋅ 2πiβ

)

and comes from the derivative of e

( )

m₁

β

. Arguing in a similar manner as above in Theorem 1 and in Lemmas 5 and 6 we show the two terms in (55) are small, which will complete the proof.

Lemma 5

(

) (

)

( )

_      =       − ₋ ≤ = ≤

∑ ∫ ∑

a m N e u du e N O N _N i _C _D N m N u u m m log , 4 2 1 1 1 2 2 β β β π . (56)

Proof. Arguing along the lines of Lemma 3, one shows the contribution from u≤ N is bounded by NlogDN. For u≥ N we apply the Siegel-Walfisz formula as in Lemma 3, giving a contribution bounded by

(16)

( )

(

( )

)

( )

 <<       −             + − ⋅             +

∫

₌ C a uN du N N O q N a c N u O q u a c N N u C q q C q log 2 log 4 φ φ β

∫

= << << N N u C C_N N du N uN log log 3β β . (57) As N N B log ≤ β , the above is _      − N N O _C _D log 2 . Lemma 6

(

) (

)

( )

_      =       − ₋ = _≤ = _≤

∫ ∑∫ ∑

N a m N e u du e mt dt O _CN _D_N t t m N u u m m 2 2 1 1 1 1 2 log , 8 1 2 2 β πβ . (58)

Proof. Arguing as in Lemma 3, one shows that the contribution when t≤ N or u≤ N is       − _N N O _C ₂_D

log . We then apply the Siegel-Walfisz Theorem as before, and find the contribution

when t,u≥ N is

∫

= = << << N N t N N u C C _N N dudt N ut log log 8 2 4 2

β

. (59) As N N D log ≤ β , the above is _      − N N O _C ₂_D 2

log . This complete the proof of Theorem 1.

We note that, for the eq. (56) and (58), the eq. (55) can be rewritten also as follows:

( )

(

) (

)

_

( )

+      − =

∑ ∫ ∑

≤ = ≤ β β β π α I i a m N e u du e N S N m N u u m m q a 1 2 2 1 1 , ; 4 ,

(

) (

)

_

( )

=      − +

_{∫ ∑∫ ∑}

= _≤ = _≤a m N e u du e mt dt N t t m N u u m m 1 1 1 1 2 1 2 2 , 8πβ β _      − _N N O _C _D log 2 + _      − _N N O _C ₂_D 2 log . (59b)

With regard the integrals over the Major arcs, we first compute the integral of u

( ) ( )

xe −x over the Major arcs and then use Theorem 1 to deduce the corresponding integral of F_N

( ) ( )

x e −x .

By Theorem 1 we know for x∈M_a_,_q_that

( )

_      <<       − − ₋ N N O q a x u a C x F_N _q _C ₂_D 2 log . (60)

(17)

We now evaluate the integral of e

( )

x q a x u _ −     

− over M_a_,_q_{; by Theorem 1 we then obtain the}

integral of F_N

( ) ( )

x e−x over M_a_,_q. Remember that

( )

∑

(

)

≤ − = N m m x m m e x u 2 1, 2 1 2 . (61) Theorem 2

∫

( )

_      +       − = − ⋅       − q a N N O N q a e d e q a u _D , 2 log M α α α . (62)

We first determine the integral of u over all of

   ₋ 2 1 , 2 1

, and then show that the integral of u

( )

x is small if N Q x > . Lemma 7

∫

( ) ( )

( )

− − = + 2 1 2 1 1 2 O N dx x e x u . (63) Proof.

∫

( ) ( )

∫ ∑ ∑

(

)

) ( )

∑ ∑ ∫

(

)

− − = − _≤ _≤ − ⋅ − = _≤ _≤ − − − 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 1 2 1 2 1 2 2 N m m N m Nm N dx x m m e dx x e x m m e dx x e x u . (64)

The integral is 1 if m1−2m2−1=0 and 0 otherwise. For m1,m2∈

{

1,...,N

}

, there are

( )

1 2 2 O N N + =    

solutions to m₁−2m₂−1=0, which completes the proof.

Define =_− + − _ N Q N Q I , 2 1 1 ,     − = N Q N Q I 2 1 , 2 . (65) The following bound is crucial in our investigations.

Lemma 8 For x∈I1 or I2,

( )

x ax e 1 1 1 _<< − for a∈

{ }

1,−2 . Lemma 9

∫

( ) ( )

∪ ∈ _     = − 2 1 I log I x D_N N O dx x e x u . (66) Proof. We have

(18)

∫

( ) ( )

∫ ∑

(

)

∫ ∑

( )

∑

(

) ( )

≤ ≤ ≤ = − ⋅ − = − − = − i i i I I N m m I m N m N dx x e x m e x m e dx x m m e dx x e x u 2 1, 1 2 2 1 2 1 2 1 2

( ) (

(

)

( )

(

)

(

)

e

( )

x dx x e x N e x e x e x N e x e i I _ −     − − + − − −       − + − =

_∫

2 1 1 2 2 1 1 (67)

because these are geometric series. By Lemma 8, we have

∫

( ) ( )

− <<

∫

<< = i i I I D_N N Q N dx x x dx x e x u log 2 2 , (68)

which completes the proof of Lemma 9. Lemma 10

∫

+

( ) ( )

(

)

− = − = N Q N Q x D N O dx x e x u 2 1 2 1 log . (69) Lemma 11

∫

( ) ( )

− _     + = − N Q N Q D N N O N dx x e x u log 2 . (70)

Proof of Theorem 2. We have

∫

( )

∫

+

( )

∫

( )

− − _ =     − − ⋅ = − ⋅       − = − ⋅       − N Q q a N Q q a N Q N Q d q a e u d e q a u d e q a u q a

β

α

, M

∫

( ) ( )

− _     +       − = −       − = N Q N Q D N N O N q a e d e u q a e log 2 β β β . (71)

Note there are two factors in Theorem 2. The first, _      − q a

e , is an arithmetical factor which depends on which Major arc M_a,_q_{we are in. The second factor is universal, and is the size of the}

contribution.

An immediate consequence of Theorem 2 is Theorem 3

∫

( ) ( )

( )

_      +       +       − = − ₋ q a N N O N N O N q a e a C dx x e x F_N _q _D _C _D , 2 log log 3 M . (72)

From Theorem 3 we immediately obtain the integral of FN

( ) ( )

x e−x over the Major arcs M :

(19)

∫

( ) ( )

∑ ∑

( )

= = = − − − =       + +       − = − N q q q a a B D C B D q N B N N N N O N q a e a C dx x e x F log 1 1 ) , ( 1 2 3 2 log log 2 M _      + + ℑ = ₋ ₋ ₋ N N N N O N B D C B D N 2 3 2 log log 2 , (73) where

∑ ∑

( )

= = =      − = ℑ N q q q a a q N B q a e a C log 1 1 ) , ( 1 (74)

is the truncated singular series for the Germain primes.

3. On some equations concerning the equivalence between the Goldbach’s Conjecture and the Generalized Riemann Hypothesis [3]

We know the Goldbach’s conjecture: “Every even integer > 2 is the sum of two primes”. In 1922 Hardy and Littlewood guesstimated, via a heuristic based on the circle method, an asymptotic for the number of representations of an even integer as the sum of two primes: Define

g 2

( )

N =#

{

p,qprime : p+q=2N

}

. Their conjecture is equivalent to g

( ) ( )

2N ≈I 2N where

( )

(

)

∫

∏

− > −       − − = 2 2 2 2 2 2 log log 2 1 2 N p N p t N t dt p p C N I (75)

and C₂, the “twin prime constant”, is defined by

(

)

∏

> =       − − = 2 2 2 1.320323... 1 1 1 2 p p C (76)

Thence, the eq. (75) can be rewritten also as follows:

( )

(

)

∏

∫

(

)

∏

− > >  −     − −       − − = 2 2 2 2 2 2 2 log log 2 1 1 1 1 2 2 N p N p p t N t dt p p p N I (77)

and thence, we obtain:

g 2

( )

N ≈

(

)

∏

∫

(

)

∏

− > >  −     − −       − − 2 2 2 2 2 2 2 log log 2 1 1 1 1 2 N p N p p t N t dt p p p , (78) or

(20)

g 2

( )

N ≈

(

)

∫

∏

− > −       − − 2 2 2 2 2 2 log log 2 1 N p N p t N t dt p p C . (78b)

We believe that a better guesstimate for g 2

( )

N is given by

( ) ( )

(

)

_             − − − =

∏

≥ ∗ 3 2 / 2 1 2 4 1 2 : 2 p p p N N N I N I . (79)

Thence, for eq. (75) we can rewrite the eq. (79) also as follows:

( )

(

)

∫

∏

− > ∗ −       − − = 2 2 2 2 2 2 log log 2 1 : 2 N p N p t N t dt p p C N I

(

)

_             − − −

∏

≥3 2 / 2 1 2 4 1 p p p N N . (79b)

Indeed it could well be that

( )

_       + = ∗ N N N O N I N g loglog log 2 2 . (80)

Thence, for eq. (79b), we obtain the following equation:

g 2

( )

N =

(

)

∫

∏

− > −       − − 2 2 2 2 2 2 log log 2 1 N p N p t N t dt p p C

(

)

_             − − −

∏

≥3 2 / 2 1 2 4 1 p p p N N         + N N N O loglog log . (80b) We introduce the function

( )

∑

= + = ) ( , 2 log log 2 prime q p N q p q p N G . (81)

The analysis of Hardy and Littlewood suggests that G 2

( )

N , plus some terms corresponding to solutions of pk +ql =2N, should be very “well-approximated” by

( )

∏

> ⋅       − − = 2 2 2 2 1 : 2 p N p N p p C N J , (82)

and the approximation g

( ) ( )

2N ≈I 2N is then deduced by partial summation. (In fact we believe that G

( ) ( )

2N =J 2N +O

(

N1/2+o( )1

)

.)

Theorem 1

The Riemann Hypothesis is equivalent to estimate

∑

(

( ) ( )

)

( ) ≤ + << − x N o x N J N G 2 1 2 / 3 2 2 . (83)

(21)

The Riemann Hypothesis for Dirichlet L-functions L

( )

s,

χ

, over all characters χmodm which are odd squarefree divisors of q , is equivalent to the estimate

(

( ) ( )

)

( ) ( )

∑

≡ ≤ + << − q N x N o x N J N G mod 2 2 2 1 2 / 3 2 2 . (84) Theorem 3

The Riemann Hypothesis for Dirichlet L-functions L

( )

s,

χ

,

χ

modq is equivalent to the conjectured estimate

∑

( ) ( ) ( )

∑

(

( )

)

≤ ≤ + + = N q x N N x o x O N G q N G 2 2 2 1 1 2 1 2

φ

. (85) Let

( )

∑

≥ +≥ = + = 3 1 , 2 log log 2 l k l k N q pk l q p N E . (86)

First note that

∑

≥+ = ≥≤ << ⋅ ≤ 3 2 3 2 2 3 / 1 2 log 1 log log log k N q p k N p l k k N N N q p , (87)

and a similar argument works for l≥3. Also it is well-known that there are o( )1

N pairs of integer q p, with p2+q2 =2N. Thus

( )

∑

(

)

= + + = N q p N N O q p N E 2 2 3 / 1 2 log log log 2 2 . (88)

Now, when we study solutions to p+q2 =2N we find that l divides p if and only if

l q

N mod

2 ≡ 2 . Thus if

(

2N/l

)

=0 or – 1 then l divides pq if and only if q≡0modl. If

(

2N/l

)

=1 then there are 2 non-zero values of q mod for which l divides p , and we also need l

to count when l divides q . Therefore our factor is 2 if l=2, and

(

)

(

)

2 / 1 1 / 2 2 1 l l l N −       ₋ + times

( ) (

)

   − −1 / 2 1 l l if _  ≥3 , 2 2 / l N l n l

Now #

{

m,n>0:m+n2 =2N

}

= 2N +O

( )

1 so we predict that

∑

∏

(

)

∏

= + ≥ >       − −       − − ≈ N q p l p N p N p p C l l N q p 2 3 2 2 2 2 2 2 1 2 / 2 1 log log , (89)

(22)

(

)

( )

∑

_∏

= + ≥ >       − −       − − ≈ prime q p N q p l p N p N N p p C l l N ) , ( 2 3 2 2 2 2 2 log 2 2 2 1 2 / 2 1 4 1 2 . (90)

Subtracting this from I 2

( )

N , we obtain the prediction I 2∗

( )

N , as in (79). We can give the more accurate prediction

( )

(

)

∫

∏

− > ∗ −       − − = 2 2 2 2 2 2 log log 2 1 2 N p N p t N t dt p p C N I

(

)

dt t N t p p N p               − +       − − −

∏

≥3 2 1 1 2 / 2 1 1 . (91)

The explicit version of the Prime Number Theorem gives a formula of the form

∑

(

)

≤ ≤ + − = x p x x O x x p ρ ρ ρ

ρ

Im 2 log log , (92)

where the sum is over zeros ρ of

ζ

( )

ρ

=0 with Re

( )

ρ

>0. In Littlewood’s famous paper he investigates the sign of

π

( )

x −Li

( )

x by a careful examination of a sum of the form

( )

∑

_ρ _ρ_≤_TLi xρ Im

: , showing that this gets bigger than

ε

−

2 / 1

x for certain values of x, and smaller than – x1/2−ε for other values of x. His method can easily be modified to show that the above implies that max log B o( )1 y p x y p y x + ≤ ≤

∑

− = (93)

where B=sup

{

Re

ρ

:

ζ

( )

ρ

=0

}

(note that 1≥B≥1/2). By partial summation it is not hard to show that

( )

(

)

(

( )

)

∑

≤ + ≤ ≤ + + + + − = = x N p q x x o B x O x x q p N G 2 Im 1 2 1 2 1 2 2 log log 2 ρ ρ ρ ρ ρ (94)

so that, by Littlewood’s method,

( )

1 ( )1 2 2 2 2 max B o y N x y x y N G + + ≤ ≤

∑

− = . (95)

Therefore the Riemann Hypothesis

(

B=1/2

)

is equivalent to the conjectured estimate

∑

( )

(

( )

)

≤ + + = x N o x O x N G 2 1 2 / 3 2 2 2 . (96)

(23)

( )

(

)

(

( )

)

( ) ( )

∑

∑ ∑

∑

∏

≤ ≤ ≤ − ≤ = − = x n n x odd d n d odd d x d n d x n d p d p n p d C p d n C n J 2 2 /2 /2 2 2 2 2 2 2 2 2 2 µ µ

( )

(

)

( )

(

x x

)

O x x O d x p d C odd d x d _p_d log 2 8 2 2 2 2 / 2 2 2 = +      + − =

∑

∏

≤ µ . (97)

Going further we note that for any coprime integers a,q≥2

( )

(

( )

)

( )

∑

≡ ≤ ≤ = +           − = q a p x p q x L qx O x a x q p mod 2 mod Im 0 , : log 1 log χ ρρχ ρ ρ

ρ

χ

φ

; (98) and thus

( )

( ) ( )1 mod log max B o q a p y p x y q x q y p + ≡ ≤ ≤

∑

−φ = , (99)

where Bq =sup

{

Reρ:L

( )

ρ,χ =0

}

for some

χ

(

modq

)

. R.C. Vaughan noted that by the same

methods but now using the above formula, we get a remarkable cancellation which leads to the explicit formula

( ) ( )

( )

(

( )

)

( ) ( ) ( )

∑

≤ ≤ ≠ ≤ = = + ₊ − = − N q x N N x q x L L qx x O x c q N G q N G 2 2 2 mod Im , Im 0 , : 0 , : 2 , 0 log 1 1 2 1 2 χ χ χ σ ρσ χ σρ ρλ σ ρ σ ρ χ φ φ (100) where c =

∫

( )

−t t −dt 1 0 1 , 1 1 ρ σ σ

ρ

_ρ

is a constant depending only on ρ and

σ

. Thus Theorem 3 follows

since ≤

( )

∫

1 − = 0 1 , 1/

ρ

1/

ρσ

σ σ ρ t dt c and has

∑

( )

≤x << qx σ

ρ

Im 2 log /

1 . Thence, the eq. (100) can be rewritten also as follows:

( ) ( )

( )

(

( )

)

( ) ( ) ( )

∑

∑ ∫

≤ ≤ ≠ ≤ = = + − _⋅ ₊ − − = − N q x N N x q x L L qx x O x dt t t q N G q N G 2 2 2 mod Im , Im 0 , : 0 , : 2 1 0 1 0 log 1 1 1 1 2 1 2 χ χ χ σ ρσ χ σρ ρλ σ ρ σ ρ

ρ

χ

φ

(100b) As in the proof of (97) we have

( )

( ) (

)

∑

≤ + = n q x n x x O q x n J 2 2 2 log 2 2 φ . (101)

(24)

∑

( ) ( )

( ) ≤ + << − x n o x n J n G 2 1 2 / 5 2 2 2 . (102)

We expect, as we saw in the precedent passages, that

( ) ( )

1/2 ( )1

2 2n J n n o G − << + and so we believe that

∑

( ) ( )

( ) ≤ + + << − x n o x n J n G 2 1 2 2 2 2 δ (103)

for δ =0. This implies, by Cauchy’s inequality, that

∑

( )

∑

( )

(

( ) ( )

)

≤ ≤ + + + + + =         + = x n n x o o x O x x O n J n G 2 2 1 2 / 3 2 1 2 3 2 / 2 2 δ δ (104)

by (97), which implies the Riemann Hypothesis if δ =0 (as after (96) above); and implies that

( )

ρ

≠0

ζ

if Reρ>3/4 if δ =1/2 (that is, assuming Hardy and Littlewood’s (102)).

We find that (85) is too delicate to obtain the Riemann Hypothesis for L

( ) (

s,

χ

,

χ

modq

)

from (103). Instead we note that

( )

(

)

( )

(

)

( ) ( ) ( )

∑

∏

≡ ≤ ≤ = + >           + − − = q N x N odd m q m primitive m x L p m p q x p m x c N G mod 2 2 2 mod Im 0 , : 1 2 2 1 2 2 2 2 2 χ χ ρρχ ρ ρ

ρ

χ

µ

(105)

plus an error term

(

2Bq o( )1

)

x O + , where

( )

(

)

(

)

( )

∏

₋− = odd p q p q p p p q q c ₂ 1 2 , 2

. As in (97) one can show that

( )

(

x x

)

O x c N J q N x N q log 2 2 mod 2 2 2 2 + =

∑

≡ ≤ , (106) so that

( ) ( )

(

1 1

)

mod 2 2 2 ) 2 2 ( C o q N x N q x O N J N G + + ≡ ≤

∑

= − (107)

where Cq =sup

{

Reρ:L

( )

ρ,χ =0

}

for some χmodm, where mq and m is odd and squarefree.

This implies Theorem 2. By the above we see that if (103) holds with δ =0 then C_q =1/2 and thus the Riemann Hypothesis follows for L-functions with squarefree conductor.

4. On some equations concerning the p-adic strings and the zeta strings [4] [5] [6] [7].

Like in the ordinary string theory, the starting point of p-adic strings is a construction of the corresponding scattering amplitudes. Recall that the ordinary crossing symmetric Veneziano