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ZEROS AND GRAM POINTS ON THE CRITICAL LINE

ζ(½±ix)

Pier Franz Roggero, Michele Nardelli1,2, Francesco Di Noto

1Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy 2 Dipartimento 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 focus attention on a relationship between zeros and Gram points with the prime numbers on the critical line ζ(½±ix)

Furthermore, we focus attention also on a formula to determine prime numbers using the Gram Points.

So if the zeros of the Riemann function give the exact number of prime numbers, with the Gram Points always on the critical line we can even find the values of all prime numbers.

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1. ZEROS AND GRAM POINTS ON THE CRITICAL LINE ζ(½ ± ix) ...................................... 3

2. OBSERVATIONS .............................................................................................................. 11

3. FORMULA TO DETERMINE PRIME NUMBERS.............................................................. 22

3. REFERENCES.................................................................................................................. 27

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1. ZEROS AND GRAM POINTS ON THE CRITICAL LINE ζ(½ ± ix)

The functional equation shows that the Riemann zeta function has zeros at −2, −4, ... . These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(πs/2) being 0 in the functional equation (see below). The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory.

The Riemann zeta function satisfies the functional equation:

( ) ( ) ( )sss

s ss −−Γ

= − 112

sin2 1 ζππζ ,

where Γ(s) is the gamma function, which is an equality of meromorphic functions valid on the whole complex plane.

It is known that any non-trivial zero lies in the open strip {s∈C: 0<Re(s)<1}, which is

called the critical strip. The Riemann hypothesis, asserts that any non-trivial zero s has

Re(s)=1/2. In the theory of the Riemann zeta function, the set {s∈C: Re(s)= 1/2} is

called the critical line.

So ζ(½ ± ix)=0

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But we also have other values on the critical line where the imaginary part IM(ζ(½±iy))=0 vanishes and they are called Gram Points.

ζ(½ ± iy)=are all transcendent real values of a combination of the function sinus.

The Riemann zeta function on the critical line can be written

( ) ( )tZeit tiθζ −=

+2

1 , ( ) ( )

+= itetZ ti

2

1ζθ

If t is a real number, then the Z function ( )tZ returns real values.

Hence the zeta function on the critical line will be real when ( )( ) 0sin =tθ . Positive real values of t where this occurs are called Gram points, after J. P. Gram, and can of

course also be described as the points where ( )π

θ t is an integer.

A Gram point is a solution ng of

( ) πθ ngn = .

Here are the smallest non negative Gram points

Gram observed that there was often exactly one zero of the zeta function between any two Gram points; Hutchinson called this observation Gram's law. There are several other closely related statements that are also sometimes called Gram's law: for example, (−1)nZ(gn) is usually positive, or Z(t) usually has opposite sign at consecutive Gram points. The imaginary parts γn of the first few zeros and the first few Gram points gn are given in the following table

g−1 γ1 g0 γ2 g1 γ3 g2 γ4 g3 γ5 g4 γ6 0.000 3.436 9.667 14.135 17.846 21.022 23.170 25.011 27.670 30.425 31.718 32.935 35.467 37.586 This shows the values of ζ(1/2+it) in the complex plane for 0 ≤ t ≤ 34. (For t=0, ζ(1/2) ≈ -1.460 corresponds to the leftmost point of the red curve.) Gram's law states that the curve usually crosses the real axis once between zeros.

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The first failure of Gram's law occurs at the 127'th zero and the Gram point g126, which are in the "wrong" order.

g124 γ126 g125 g126 γ127 γ128 g127 γ129 g128 279.148 279.229 280.802 282.455 282.465 283.211 284.104 284.836 285.752

A Gram point t is called good if the zeta function is positive at 1/2 + it.

The indices of the "bad" Gram points where Z has the "wrong" sign are 126, 134, 195, 211, ... A Gram block is an interval bounded by two good Gram points such that all the Gram points between them are bad. A refinement of Gram's law called Rosser's rule due to Rosser, Yohe & Schoenfeld (1969) says that Gram blocks often have the expected number of zeros in them (the same as the number of Gram intervals), even though some of the individual Gram intervals in the block may not have exactly one zero in them. For example, the interval bounded by g125 and g127 is a Gram block containing a unique bad Gram point g126, and contains the expected number 2 of zeros although neither of its two Gram intervals contains a unique zero. Rosser et al. checked that there were no exceptions to Rosser's rule in the first 3 million zeros, although there are infinitely many exceptions to Rosser's rule over the entire zeta function.

Gram's rule and Rosser's rule both say that in some sense zeros do not stray too far from their expected positions. The distance of a zero from its expected position is controlled by the function S defined above, which grows extremely slowly: its average value is of the order of (log log T)1/2, which only reaches 2 for T around 1024. This means that both rules hold most of the time for small T but eventually break down often. Indeed Trudgian (2011) showed that both Gram's law and Rosser's rule fail in a positive proportion of cases.

To be more specific, it is expected that in about 73% one zero is enclosed by two successive Gram points, but in 14% no zero and in 13% two zeros are in such a Gram-interval on the long run.

Hence the function ζ(½ + ix) has others values where the IM(ζ(½ + ix))=0 and so we find the correlation with the Gram points g IM(ζ(½ + ig))=0

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As we can see from this graph

Blue is the real part and red is the imaginary part of the function ζ(½ ± ix) for 0≤x≤100 is shown so that we can clearly see the first non-trivial zeros and the values where real and imaginary part are both equal to zero: Re(ζ(½±ix)) = IM(ζ(½±ix)) = 0

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Combining the zeros and points of Gram we have the following table

TAB. 1

x jy prime number

0 -1,4603545

3,436218226 0,56415 2, 3 9,666908056 1,53181 5, 7 14,13472514 0 11, 13 17,845599 2,34018 17 21,02203964 0 19 23,1702827 1,45744 23 25,01085758 0 27,67018222 2,84509 30,42487613 0 29 31,71797995 0,925264 31 32,93506159 0 35,4671843 2,93812 37,58617816 0 37 38,99920996 1,786721 40,91871901 0 42,36355039 1,098756 41 43,32707328 0 43 45,59302898 3,6629 48,00515088 0 47 48,71077662 0,688292 49,77383248 0 51,73384281 2,0112138 52,97032148 0 54,67523745 2,91239 53 56,4462477 0 57,54516518 1,758164

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59,347044 0 59 60,35181197 0,53858 60,83177853 0 63,10186798 4,1643988 61 65,11254405 0 65,80088764 1,053877 67,07981053 0 67 68,45354492 1,54 69,54640171 0 71,05 1,96 71 72,06715767 0 73,64 3,61 73 75,7046907 0 76,15 0,55 77,14484007 0 78,65 1,24 79,33737502 0 79 81,1 3,99 82,91038085 0 83,58 1,16 83 84,73549298 0 85,99 1,94 87,42527461 0 88,38 0,64 88,80911121 0 90,75 4,47 89 92,49189927 0 93,09 1,3 94,65134404 0 95,4 0,49 95,87063423 0 97,7 2,86 97 98,83119422 0 99.99 2,69 101,317851 0 101 102,25 2,12

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103,725538040 0 103 104,49 1,01 105,446623052 0 106,72 0,98 107,168611184 0 107 108,93 5,18 111,029535543 0 109 111,1 0,1 111,874659177 0 113,3 1,78 113 114,320220915 0 115,4 1,59 116,226680321 0 117,6 3,09 118,790782866 0 119,7 2,64 121,370125002 0 121,8 0,73 122,946829294 0 124 0,47 124,256818554 0 126,1 4,56 127,516683880 0 127

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We have intervals that can be of all 4 types: between a zero and a Gram point or between a Gram point and a zero or between 2 zeros or still between two Gram points: In summary: ZG GZ ZZ GG The prime numbers starting with number 47 are all spaced out by at least one interval. The statement is as follows: Prime numbers, starting with number 47, are all within any one of these intervals (ZG, GZ, ZZ or GG), however, two consecutive primes are never on two contiguous intervals, which is equivalent to say that also two consecutive primes or twin prime are spaced by at least one interval. Thus for example the odd number 91 is not a prime number, in the range between 90,8 to 92,49189927 because the interval immediately preceding already exists the prime number 89. Another example, where G and Z are not alternating, are the twin prime numbers 281 and 283 280.802 282.455 281 282.465 283.211 283 In bold there are the two zeros We note that the prime twin numbers 281 and 283 are separated by exactly one interval.

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2. OBSERVATIONS Since the Gram points are involved both π = 3.14 and the prime numbers of the form 6k +1 except 2 and the 3 initials, and since 6 ≈ 2π = 6.28, many of the primes involved are also very close to 2πk + 1 if the number is even, 2πk + 1 if the number is odd, or with the integer part of 2πk. For example in the following table below: numbers first column. TAB 1

2πk + 1 2πk + 2 2πk integer part first = prime involved in the numbers of Gram

14,13472514 14 pari

13 and 15

23,1702827 23 dispari

21 and 25 23

31,71797995 ... 29 and 33 31 37,58617816 35 and 39 37 43,32707328 41 and 45 43 52,97032148 51 and 53 59,347044 57 and 61 59 60,83177853 59 and 61 Third column TAB. 1 with prime numbers (in red) and common involved in the above table 2, 3 5, 7 11, 13 17

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19 23 29 31 37 41 43 47 43 47 53 59 61 ...

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Subsequent reports of the numbers of Gram, for helpful comments on the performance of than that of the primes of similar size Indices n Gram points g(n)

Subsequent relations r n/(n-1)

Primes of the same size from 127

Subsequent relations r’ p/(pn -1)

comments The ratio decreases more and more and tends to 1 as n increases, even for prime numbers r decreases faster than r '

126 134/126 = 1,06 127 131/127=1,03 134 195/134 = 1.45 131 ...= 1,04 195 211/195= 1,08 137 ...=1,01 137 e 139

are twins

211 232/211=1,0995 139 ...= 1,07 232 254/232=1,0948 149 ...1,01 151 e 149

are twins

254 288/254=1,1338 151 ...=1,039 288 367/288=1,2743 157 ...=1,038 367 377/367=1,0272 163 …=1,024 377 379/377=1,0053 167 …=1,035 379 173

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The relationships are almost similar, both tend to 1 to grow it p. Relationships are about small r the greatest powers of r 'corresponding to the primes, for example p is 1.45, we have that the corresponding ratio is 1.04 so 1,049 = 1,42 ≈ 1,45, or 1,0383 = 1,12 ≈1,1338 , but also 1,0387 = 1,29 ≈1,2743 Indices n of Gram points g(n) for which (-1)^n Z(g(n)) < 0, where Z(t) is the

Riemann-Siegel Z-function.

126, 134, 195, 211, 232, 254, 288, 367, 377, 379, 397, 400, 461, 507, 518, 529, 567, 578, 595, 618, 626, 637, 654, 668, 692, 694, 703, 715, 728, 766, 777, 793, 795, 807, 819, 848, 857, 869, 887, 964, 992, 995, 1016, 1028, 1034, 1043, 1046, 1071, 1086

Prime numbers from 127 to 277 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 On some equations concerning the Gram’s Law in the theory of Riemann Zeta function Definition 1 For positive t , we denote by ( )tϑ the increment of the argument of the function

Γ−

22/ ssπ along the segment with the end-points

2

1=s and its +=2

1 . On can prove that

the asymptotic expansion

( ) ( )( ) ( )∑+∞

=

−−+−

−−

−+−−≈1

122

1

2

12

11222

12

822ln

2 n

nn

n

n

n

tBnn

tttt

ππ

ϑ (1)

holds, as t grows. Here the nB2 are the Bernoulli numbers. The function ( )tϑ is presented in the expression for ( )tN – the number of zeros of ( )sζ in the strip ts≤< Im0 .

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This expression is called by Riemann – von Mangoldt formula and has the form

( ) ( ) ( )tSttN ++= 11ϑπ

. (2)

Here ( )

+= − ittS2

1arg1 ζπ denotes the argument of the Riemann zeta function on the

critical line. Thence, the eq. (2) can be rewritten also as follows:

( ) ( ) ++= 11

ttN ϑπ

+− it2

1arg1 ζπ . (2b)

Definition 2 For any 0≥n , the Gram point nt is defined as the unique root of the equation

( ) ( )1−⋅= ntn πϑ . Definition 3 Suppose ba, are an arbitrary real numbers, ba < . Denote by ( )bae , the number of solutions of the inequalities ba n ≤∆< with the condition MNnN +≤< . Similarly, by

( )baf , we denote the number of solutions of the inequalities ( ) bna ≤∆< under the same condition. Lemma 1 The relation ( ) ( ) ( )( ) ( )21,1, ++++−+−= baabfbae θ (3)

holds true for any integers a and ,b ba <

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Lemma 2 For a real x let the quantity ( )xν denote the number of integers n , MNnN +≤< , satisfying the condition

( )2

Lxn

π≤∆ . (4)

Then

( )

+= ∫ ∞−

−x u dueMx θδπ

ν 2/2

2

1 . (5)

where ( ) 5.05.14.22 ln −−= Le εδ . Thence, we can rewrite the eq. (5) also as follows:

( ) ( )( )

+= ∫ ∞−

−−−x u LedueMx 5.05.14.222/ ln2

1 2

εθπ

ν . (5b)

We note that the values 22.4, 1.5 and 0.5 can be obtained also as follows: 22.4 = 537.6/24; 1.5 = 36/24; 0.5 = 12/24 Thence, we can rewrite the eq. (5b) also as follows:

( ) ( )( )

+= ∫ ∞−

−−−x u LedueMx 24/1224/3624/6.5372/ ln2

1 2

εθπ

ν . (5c)

In this way, we can to connect the eq. (5c) with the number 24 that is related to the “modes” that correspond to the physical vibrations of the bosonic strings by the following Ramanujan function:

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( )

++

+

⋅

=

−

∞ −∫

42710

421110

log

'142

'

cosh'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

Thence, we have the following mathematical connection:

( ) ( )( ) ⇒

+= ∫ ∞−

−−−x u LedueMx 24/1224/3624/6.5372/ ln2

1 2

εθπ

ν

( )

++

+

⋅

⇒

−

∞ −∫

4

2710

4

21110log

'

142

'

cosh

'cos

log42

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

. (5d)

Theorem 1 For any a and b , ba < , the number of solutions of the inequality ba n ≤∆< with the condition MNnN +≤< satisfies the relation

( )

∆+= ∫−β

αθ

πdueMbae u 2/2

2

1, , (6)

where La /2πα = , Lb /2πβ = , ( ) 5.05.14.22 ln2.2 −−=∆ Le ε . Also here, as for the eqs. (5b-5c), we can rewrite the eq. (6) also as follows:

( ) ( )( )

+= ∫−−−Lb

La

u LedueMbae/2

/2

24/1224/3624/6.5372/ ln2.22

1,

2π

πεθ

π, (6b)

that can be connected with the Ramanujan’s function concerning the number 24:

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( ) ( )( ) ⇒

+= ∫−−−Lb

La

u LedueMbae/2

/2

24/1224/3624/6.5372/ ln2.22

1,

2π

πεθ

π

( )

++

+

⋅

⇒

−

∞ −∫

4

2710

4

21110log

'

142

'

cosh

'cos

log42

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

. (6c)

Let ba, be an integers and let c be sufficiently large constant such that the inequalities

ln ≤∆ , ( ) ln ≤∆ hold true for any MNnN +≤< with [ ]Ncl ln= . Then in the case

lbal ≤<≤− the assertion follows from lemmas 1 and 2:

( ) ( ) ( )( )( )

( )=+

+=++−+−= ∫+−

+−

− ldueMlabfbaeLa

Lb

u1

/21

/21 22/

1 222

121,1,

2

θδθπ

θπ

π

+=+

++= ∫∫−− β

α

π

πθδ

πθδθπθ

π1.2

2

122/

2

1 2/12

/2

/2 22/ 22

dueMlLdueM uLb

La

u . (7)

In the case blal <<≤− the required statement follows from the equality ( ) ( )laebae ,, = and from the estimate

∫ << −−−β

λλ δλ

π01.0

2

1 2/12/ 22

edue u , Ll /2πλ = . (8)

The cases lbla ≤≤−< , blla <<−< are handle as above. If a or b is non-integer, then assertion follows from the relation ( ) [ ] [ ]( )baebae ,, = and the above arguments. Theorem 1 asserts that the quantity ( )bar , in the relation

( ) ( )

+= ∫−β

απbardueMbae u ,

2

1, 2/2

(9)

obeys the estimate ( ) ( )( ) ( )( )5.05.0 lnlnlnln, −− == NOLObar , (10)

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for any a and b . Suppose ε be any positive number, 2

1<ε . Then for any a we have

( ) ( )

−+++=−++= ∫−2

1

2

1,2

11,0 2/α

αεε

πεε aardueMaae u , (11)

where ( ) La /21 επα += , ( ) La /212 επα −+= . Hence,

( ) ( ) ( ) ( )( )∫−−− ==−=−=−++ 2

1

2 5.05.012

2/ lnln2

11,

α

ααα

πεε NOLOOdueaar u . (12)

This equation can be rewritten also as follows:

( ) ( ) ( ) ( )( )∫−−− ==−=−=−++ 2

1

2 24/1224/1212

2/ lnln2

11,

α

ααα

πεε NOLOOdueaar u . (12b)

Also the equation (12b), thence, can be related with the Ramanujan’s equation concerning the number 24:

( ) ( ) ( ) ( )( )⇒==−=−=−++ ∫−−−2

1

2 24/1224/1212

2/ lnln2

11,

α

ααα

πεε NOLOOdueaar u

( )

++

+

⋅

⇒

−

∞ −∫

4

2710

4

21110log

'

142

'

cosh

'cos

log42

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

. (12c)

Probably this estimate holds true for any ba, . Theorem 2 The quantity ( )xν of the numbers MNnNn +≤<, that satisfy the condition ( ) ( ) xLtte Nnnn ≤−= /2'ϑγ (13)

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obeys the following relation:

( )

+= ∫ ∞−

−x u

L

AdueMx

ln2

1 2/2 θπ

ν . (14)

Suppose ( )tf be a characteristic function of the discrete random quantity with the values

MNnNen +≤<, , that is, the sum

( ) ( )∑+≤<

=MNnN

niteM

tf exp1 . (15)

Taking an integer 1>K whose precise value is chosen below, we get

( ) ( ) ( )( ) ( )( ) ( ) ( )

( ) ( ) ( )( )∑ ∑ ∑ ∑

+≤< +≤<

−

= =

−−

=

+

−−+−=+=

MNnN MNnN

K

k

K

k

Knk

n

kk

n

k

nn K

tete

kite

kMteite

Mtf

1

0 1

212

12

!22

!12

1

!2

11sincos

1 θ

( )( )

( )( ) ( )∑ ∑ ∑ ∑ ∑

−

= +≤< = +≤< +≤<

−−−

=+−

−+−=1

0 1

22

12121

22 1

!22

!12

11

!2

1K

k MNnN

K

k MNnN MNnN

Kn

Kk

n

kkk

n

kk

eMK

te

k

tie

Mk

t θ

( )( )

( ) ( )( )

( ) ( )( )

( ) ( )∑ ∑

= =

−

++−

−+

+−=

K

k

K

k

K

K

Kkkk

k

kk

L

A

K

K

K

t

L

Be

Bk

k

k

t

L

A

k

k

k

t

0 1

62

3

212

2

61

2 342.01

!2

!2

!22

6.1

!

!12

!12

342.01

!2

!2

!2

1 θπθθ

(16) where 01 =θ for 0=k . After some transformations we have

( ) ( ) ( ) ( )

++

+= ∑ ∑

=

−

=

−−K

k

kK

k

kk

kK

t tk

Be

BL

tt

k

A

L

tt

Ketf

1

12

1

212

622/

!

4.2

!

324.0

2!

32 πθ . (17)

Now let’s consider the integral ( )λI ,

( ) ( ) ( )∫

−=

λλ

0dt

t

tgtfI , (18)

where 1>λ . Thence, we have the following estimate:

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( ) ( ) ( ) ( )++

<

−++

≤ ∑ ∑

= =

−26

2

1 1

122262

32exp2.0

212!4.2

2!324.0

2!13 λλλπλλλ A

LK

e

kk

Be

BLkk

A

LKKI

KK

k

K

k

kkkkK

( ) ( )222

22 exp5.0

2exp

4.2 λπλλπ BeLK

eBe

BL

K

+

<+ . (19)

Setting

Be

Lln

2

1

πλ = , 1ln

8

11

2 22 +

=+

= LBe

Kπ

λ , (20)

we get:

( )4

22 5.0exp

5.0

LBe

L<λπ ,

LLL

BB

K

e K

K1

ln5

expln8ln

exp2 22

2

<

−<

−≤≤

−

ππλ . (21)

Thus, ( ) 4 LI <λ . Let ( )xF and ( )xG be the distribution functions corresponding to the characteristic functions ( )tf and ( )tg , respectively. By the Berry-Esseen inequality, for any real x we have:

( ) ( ) ( )L

A

L

e

LIxGxF

lnln

12122 5.15.20

4

1 <+<+≤−−

− ελππ

λπ

. (22)

Hence,

( ) ( ) ( ) ∫ ∞−

−− +=++=x u

L

AdueIxGxF

ln2

12122 2/1 2 θπ

λππ

λπ

, (23)

or

( ) ( ) ( ) ∫ ∞−

−−

+=+=++=x u

L

Adue

L

AxG

L

e

LxGxF

ln2

1

lnln

1 2/5.15.20

4

2 θπ

θε . (23b)

We note that the eq. (23) can be rewritten also as follows:

( ) ( ) ( ) ∫ ∞−

−− +=++=x u

L

AdueIxGxF

ln

2

2

222442 2/1 2 θ

πλ

ππλ

π, (23c)

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where the number 24 is related to the “modes” that correspond to the physical vibrations of the bosonic strings by the following Ramanujan function:

( )

++

+

⋅

=

−

∞ −∫

42710

421110

log

'142

'

cosh'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

While, the eq. (23b) can be rewritten also as follows:

( ) ( ) ( ) ∫ ∞−

−−

+=+=++=x u

L

Adue

L

AxG

L

e

LxGxF

ln2

1

lnln

1 2/16/2424/492

4

2 θπ

θε , (23d)

where also here there is the number 24, fundamental in string theory.

3. FORMULA TO DETERMINE PRIME NUMBERS The formula to determine the prime number pn is the following:

( )( )1

1

ln −

−

−+⋅=

nn

nnn xx

xxp

ε

with 0 < ε < 1 Depending on the value of ε we find are all the prime numbers except the primes 2 and 3. Of course, choosing the appropriate value of ε we have the prime number in integer value. For example if we choose ε= 0,45156362…. we have the prime integer number 41.

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As we can see in TAB. 1 with ε=0,57 for all values we have a good approximation for all prime numbers up to 193:

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TAB. 1

x of Gram points value

prime

number

calculated

with

ε=0,57

prime

number

0 -1,4603545 2

3,43621823 0,56415 3

9,66690806 1,53181 4,082445932 5

17,845599 2,34018 7,462236229 7

23,1702827 1,45744 13,97973288 11

27,6701822 2,84509 19,2672899 13

31,71798 0,925264 24,21106208 17

35,4671843 2,93812 28,97788798 19

38,99921 1,786721 33,63721509 23

42,3635504 1,098756 38,22580999 29

45,593029 3,6629 42,76581517 31

48,7107766 0,688292 47,27170627 37

51,7338428 2,0112138 51,75350473 41

54,6752374 2,91239 56,21844895 43

57,5451652 1,758164 60,67194099 47

60,351812 0,53858 65,11811857 53

63,101868 4,1643988 69,5602188 59

65,8008876 1,053877 74,0008189 61

68,4535449 1,54 78,44200124 67

71,05 1,96 83,33832578 71

73,64 3,61 86,66276201 73

76,15 0,55 92,77616005 79

78,65 1,24 96,29694695 83

81,1 3,99 101,6166911 89

83,58 1,16 103,3489847 97

85,99 1,94 109,88172 101

88,38 0,64 114,07283 103

90,75 4,47 118,3280665 107

93,09 1,3 123,2590548 109

95,4 0,49 128,3244164 113

97,7 2,86 132,1476701 127

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99,99 2,69 136,0003051 131

102,25 2,12 141,3806411 137

104,49 1,01 146,1194377 139

106,72 0,98 150,1115486 149

108,93 5,18 155,0083987 151

111,1 0,1 161,8855066 157

113,3 1,78 162,2256401 163

115,4 1,59 175,7008202 167

117,6 3,09 168,4428438 173

119,7 2,64 182,3078471 179

121,8 0,73 185,5345347 181

124 0,47 177,6963563 191

126,1 4,56 192,1415616 193

For example, we have that: 126,1 + 124 = 250,1; 250,1 * 0,57 = 142,557 ; 126,1 – 124 = 2,1; ln 2,1 = 0,741937344; 142,557 / 0,741937344 = 192,1415618 thence the above formula is correct. Furthermore, all the numbers of TAB 1. are connected with the following numbers concerning the universal music system based on Phi, thence we have some mathematical connections also with π and Φ : 0,5648; 1,5326; 2,34375; 1,456230; 2,8328; 0,92705; 2,9361; 1,78329; 1,09872; 3,6668; 0,6875; 2,0162; 2,91246; 1,7507; 0,53498; 4,1666; 1,0524.

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With regard the mathematical connections with the Ramanujan’s modular equations regarding the physical vibrations of the bosonic strings that are connected also with

π (thence with Φ by the simple expression Φ=⋅65π ), we have that:

−−++

+=−==

∫q

t

dt

tf

tfqR

0 5/45/1

5

)(

)(

5

1exp

2

531

5)(

2

15/1618033,0 φ ,

and

−−++

+−Φ=

∫q

t

dt

tf

tfqR

0 5/45/1

5

)(

)(

5

1exp

2

531

5)(

20

32π ,

where 2

15 +=Φ .

Furthermore, we have also:

++

+=4

27104

21110log

142

24π .

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3. REFERENCES

Aggirandosi tra i plot della zeta di Riemann - Rudi Mathematici www.rudimathematici.com/blocknotes/pdf/RTGFZ.pdf Lo studio della zeta di Riemann è interessante non solo da un punto di vista .... L'ipotesi di Riemann è uno dei problemi del Millennio non ancora risolti. ... come indicatori i punti di Gram oppure i valori di minimo o massimo locale della Z(t), .. References in Google “gram points Riemann” Riemann–Siegel theta function - Wikipedia, the free encyclopedia en.wikipedia.org/.../Riemann–Siegel_theta_functio... Traduci questa pagina Passa a Gram points - [edit]. The Riemann zeta function on the critical line can be written. \zeta\left(\frac{1}{2}+it\: Z(t) = e^{i \theta(t)} \. If t is a real number, ... Riemann hypothesis - Wikipedia, the free encyclopedia en.wikipedia.org/wiki/Riemann_hypothesis Traduci questa pagina Passa a Gram points - [edit]. A Gram point is a point on the critical line 1/2 + it where the zeta function is real and non-zero. Using the expression for the ... Gram Point -- from Wolfram MathWorld mathworld.wolfram.com/GramPoint.html Traduci questa pagina Gram Point. DOWNLOAD Mathematica Notebook GramPoints. Let theta(t) be the Riemann-Siegel function. The unique value g_n such that ...

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number theory - Why are Gram points for the Riemann zeta ... math.stackexchange.com/.../why-are-gram-points-f... Traduci questa pagina 12/nov/2011 - Given the Riemann-Siegel function, why are the Gram points important? I say if we have $S(T)$, the oscillating part of the zeros, then given a ... [PDF] On the values of the Riemann zeta function at the Gram points www.mccme.ru/lifr/2011global/talks/korolev.pdf Traduci questa pagina On the values of the Riemann zeta function at the Gram points. M.A. Korolev. The Gram point tn > 0 in the theory of ζ(s) may be interpreted as the value of ... Riemann Hypothesis in a Nutshell - Home Page web.viu.ca/.../Riemannzeta/riemannzetalong.html Traduci questa pagina Riemann Hypothesis. ... This alternation of zeros with Gram points is key to verifying the Riemann Hypothesis. The algorithm used to compute Z(t) is called the ... [PDF] Gram's Law and the Argument of the Riemann Zeta Function elib.mi.sanu.ac.rs/files/journals/.../n106p053.pdf Traduci questa pagina di M Korolev - Citato da 2 - Articoli correlati the Riemann zeta function at the Gram points are proved. We apply these ... The notion 'Gram's Law' has different senses in different papers. Thus, we begin this ... [PDF] Separation of the complex zeros of the Riemann zeta function www.mathematik.hu-berlin.de/.../Herman%20J.J.%... Traduci questa pagina 20/lug/2006 - The Euler-Maclaurin formule voor абг ¢. ¤. The Riemann-Siegel formula for ебз жд. Gram points and Gram's “Law”. Rosser's rule. Separation of ... Mean values of the Riemann zeta-function at the Gram points ... www.maths.ox.ac.uk/node/9370 Traduci questa pagina 21/mag/2009 - Mean values of the Riemann zeta-function at the Gram points. Mean values of the Riemann zeta-function at the Gram points. Thu, 21/05/2009

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On the Gram's Law in the Theory of Riemann Zeta Function ... www.researchgate.net/.../47817386_On_the_Gram... Traduci questa pagina Function and the Riemann Hypothesis' (1946), and some of their equivalents. 0 0 ... For any n?0, the Gram point tn is defined as the unique root of the equation ...