riemann hypothesis from area quantization, renormalization group

8/14/2019 Riemann Hypothesis from Area Quantization, Renormalization Group

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On the Riemann Hypothesis, AreaQuantization, Dirac Operators,

Modularity and Renormalization Group

Carlos CastroCenter for Theoretical Studies of Physical Systems

Clark Atlanta University, Atlanta, GA. 30314, [email protected]

8 December 2008, Revised December 17, 2008

Abstract

Two methods to prove the Riemann Hypothesis are presented. One isbased on the modular properties of (theta) functions and the other onthe Hilbert-Polya proposal to nd an operator whose spectrum reproducesthe ordinates n (imaginary parts) of the zeta zeros in the critical line :sn = 12 + in . A detailed analysis of a one-dimensional Dirac-like operatorwith a potential V (x) is given that reproduces the spectrum of energy lev-els E n = n , when the boundary conditions E (x = ) = E (x =+ ) are imposed. Such potential V (x) is derived implicitly from therelation x = x(V ) = 2 (d N (V )/dV ), where the functional form of N (V )is given by the full-edged Riemann-von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the O (E

n ) terms.

The construction is also extended to self-adjoint Schroedinger operators.Crucial is the introduction of an energy-dependent cut-off function ( E ).Finally, the natural quantization of the phase space areas (associated tononperiodic crystal-like structures) in integer multiples of follows fromthe Bohr-Sommerfeld quantization conditions of Quantum Mechanics. Itallows to nd a physical reasoning why the average density of the primesdistribution for very large x (O ( 1logx )) has a one-to-one correspondencewith the asymptotic limit of the inverse average density of the zeta zerosin the critical line suggesting intriguing connections to the Renormaliza-tion Group program.

Keywords: Quantum Mechanics, Dirac Operators, Riemann Hypothesis, Hilbert-

Polya conjecture, Area Quantization, Modularity, Renormalization Group, StringTheory.

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1 Introduction : Riemann Hypothesis, Scaling

and Modular Invariance

Riemanns outstanding hypothesis [1] that the non-trivial complex zeros of thezeta-function (s) must be of the form sn = 1 / 2in , is one of most importantopen problems in pure mathematics. The zeta-function has a relation with thenumber of prime numbers less than a given quantity and the zeros of zeta aredeeply connected with the distribution of primes [1]. References [2] are devotedto the mathematical properties of the zeta-function.

The RH has also been studied from the point of view of mathematics andphysics [6], [9], [14], [18], [11], [19] among many others. A novel physical inter-pretation of the location of the nontrivial Riemann zeta zeros which correspondsto the presence of tachyonic-resonances/tachyonic-condensates in bosonic string

theory was found in [7] : if there were zeros outside the critical line violating theRH these zeros do not correspond to any poles of the string scattering amplitude.The spectral properties of the n s are associated with the random statisticaluctuations of the energy levels (quantum chaos) of a classical chaotic system[14]. Montgomery [8] has shown that the two-level correlation function of thedistribution of the n s coincides with the expression obtained by Dyson withthe help of random matrices corresponding to a Gaussian unitary ensemble.

In [10] by constructing of a continuous family of scaling-like operators involv-ing the Gauss-Jacobi theta series and logarithmic derivatives, and after invokinga CT -symmetry corresponding to a judicious charge conjugation Cand time re-versal T operation, we were able to show that the Riemann Hypothesis follows.The charge conjugation operation Cis related to scalings transformations, andtime reversal

T operation, is related to the inversions t

(1/t ) such that

log(t) log(t). A Wick rotation of variables t = iz furnishes z (1/z )which is a modular SL (2, Z ) transformation z (az + b/cz + d) with unitdeterminant ad bc = 1.For these reasons, before entering into the next two sections we deem itvery important to review the results [10], [16] based on a family of scaling-like operators in one dimension involving the Gauss-Jacobi theta series andan innite parameter family of theta series where the inner product of theireigenfunctions s (t; l) is given by (2 /l )Z [2l (2k ss)], where Z (s) is theRiemann completed zeta function and the l, k parameters are constrained toobey (l + 4) / 8 = k in order to have CT -invariance.There is a one-to-one correspondence among the zeta zeros sn ( Z [sn ] = 0 (sn ) = 0 ) with the eigenfunctions s n (t; l) (of the latter scaling-like operators)when the latter are orthogonal to the ground reference state s o (t; l); whereso = 12 + i0 is the center of symmetry of the location of the nontrivial zeta zeros. We shall present a concise review [10] and show why the Riemann Hypothesisfollows from a CT invariance when the pseudo-norm of the eigenfunctions is not null . Had the pseudo-norm < s |CT|s > been null , theRH would have been false.

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The Scaling Operators related to the Gauss-Jacobi Theta series and theRiemann zeros [16] are given by

D1 = d

d ln t+

dV d ln t

+ k. (1.1)

such that its eigenvalues s are complex-valued, and its eigenfunctions are givenby

s (t) = ts + k eV ( t ) . (1.2a)D1 is not self-adjoint since it is an operator that does not admit an adjointextension to the whole real line characterized by the real variable t. The pa-rameter k is also real-valued. The eigenvalues of D1 are complex valued numberss. The charge conjugation operation Cacting on the eigenfunctions is denedas

s (t) = ts + k eV ( t ) s (t) = ts + k eV ( t ) =

ts

+ s s (t). (1.2b)

which is related to scalings transformations of s (t) by t-dependent (local) scal-ing factors

ts + s = e(s

+ s ) ln t = e 2 i I m (s ) lnt a phase rotation (1.2c)

where I m(s) is the imaginary part of s. Since local t-dependent ( lnt dependentto be precise ) phase rotations resemble U (1) gauge transformations one canthen interpret the ( dV/dlnt ) term in D1 as a gauge eld (potential) in one-dimension that gauges the scalings transformations. V is the pre-potential andA = ( dV/dlnt ) is the potential. Thus charge conjugations (1.2b) can be recast

as scaling transformations (1.2c).We also dene the mirror operator to D1 as follows,

D2 =d

d ln t dV (1/t )

d ln t+ k. (1.3)

that is related to D1 by the substitution t 1/t and by noticing thatdV (1/t )d ln(1/t )

= dV (1/t )

d ln t. (1.4)

where V (1/t ) is not equal to V (t) and D2 is not self-adjoint either. Whenl = 4(2 k 1), the eigenfunctions of the D2 operator are s ( 1t ) (with eigenvalues), and which can be shown to be equal to 1s (t) [16]. This results from theproperties of the Gauss-Jacobi theta series under the x 1/x transformations.Since V (t) can be chosen arbitrarily, we chose it to be related to the Bernoullistring spectral counting function, given by the Jacobi theta series,

e2V ( t ) =

n = en 2 t l = 2 (t l ) + 1 . (1.5)

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this is where the l parameter appears in (1.5); the k parameter appears in (1.1,1.3). The condition l = 4(2 k

1) [16] is required so the orthogonal states s n (t)

(parametrized by the complex eigenvalues sn ) to the ground state s =1 / 2(t)have a one-to-one correspondence to the zeta zeros zn in such a way that thequartets of numbers {sn }are symmetrically located w.r.t the critical line, andreal axis, in the same way that the zeta zeros zn are : the quartets are {sn }=sn ; 1 sn ; sn ; 1 sn . Furthermore the condition l = 4(2 k 1) is required inorder construct CT -invariant (but not Hermitian) Hamiltonians as we describebelow.

The related theta function dened by Gauss was

G(1/x ) =

n = en 2 /x = 2 (1/x ) + 1 . (1.6)

where (x) =

n =1 en 2 x

. The Gauss-Jacobi series obeys the relation

G(1x

) = x G(x). (1.7)resulting from the Poisson re-summation formula. The V (t) is dened as e2V ( t ) =G(t l ) where x = t l . The pair of mirror Hamiltonians H A = D2D1 and H B =D1D2 , when l = 4(2 k 1) obey

H A s (t) = s(1 s)s (t). H B s (1t

) = s(1 s) s (1t

). (1.8)

due to the relation s (1/t ) = 1s (t) based on the modular properties of theGauss-Jacobi series, G( 1x ) = x G (x). Therefore, despite that H A , H B are notHermitian they have the same spectrum s(1 s) which is real -valued only inthe critical line and in the real line. Eq-(1.8) is the one-dimensional version of the eigenfunctions of the two-dimensional hyperbolic Laplacian given in termsof the Eisensteins series.

Had H A , H B been Hermitian one would have had an immediate proof of theRH. Hermitian operators have a real spectrum, hence if s(1 s) is real thismeans that s = 12 + i, and/or s = real . The trivial zeta zeros are locatedat the negative even integers (real) and the nontrivial zeta zeros are locatedin the critical line s = 12 + i. From eq-(1.8) and using the properties of theGauss-Jacobi series G( 1x ) = x G(x) it follows that under the time reversal T operation t 1t the eigenfunctions s (t) behave as

T s (t) = s (1t

) = 1s (t). (1.9)

such that the Hamiltonian operators H A = D2D1 , H B = D1D2 transform as

T H B T 1 = H A , T H A T 1 = H B . (1.10a)the combined action of CT transformations is implemented on the states asfollows

C T H A [ C T ]1 s (t) = H A s (t)

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C T H B [ C T ]1 s (t) = H B s (t). (1.10b)since s (t) span a continuum of eigenfunctions, for a continuum of s values, eqs-(1.10b) result in the vanishing of the commutators [ H A , CT ] = [H B , CT ] = 0.When the operators H A , H B commute with CT , there exits new eigenfunctionsCT s (t) of the H A operator with eigenvalues s(1 s). Let focus only in theH A operator since similar results follow for the H B operator. Dening

| CT s (t) > CT |s (t) > . (1.11)one can see that it is also an eigenfunction of H A with eigenvalue s(1 s) :

H A | CT s (t) > = H A CT |s (t) > = H A | 1s (t) > =s(1

s)

|1

s (t) > = s(1

s)

CT |s (t) > = ( E s )

|CT s (t) > .

(1.12)where we have dened ( E s )= s(1 s). The CT action on s(1 s) s isdened to be linear : s(1 s) CT s since Cacts only on the states s (t) asscalings (1.2b), and not on the numbers s(1 s). Therefore, one has

[H A , CT ] = 0 < s | [H A , CT ] | s > = 0 < s | H A CT |s > < s | CT H A | s > =

(E s )< s | CT |s > E s < s | CT |s > =(E s E s ) < s | CT |s > = 0 . (1.13)

Similar results follow for the H B operator. From (1.13) one has two cases toconsider.

Case A : If the pseudo-norm is null< s | CT |s > = 0 (E s E s ) = 0 (1 .14)

then the complex eigenvalues E s = s(1 s) and E s = s(1 s) are complexconjugates of each other. In this case the RH would be false and there arequartets of non-trivial Riemann zeta zeros given by sn , 1 sn , sn , 1 sn .

Case B : If the pseudo-norm is not null :< s | CT |s > = 0 (E s E s ) = 0 (1 .15)

then the eigenvalues are real given by E s = s(1 s) = E s = s(1 s) andwhich implies that s = real ( location of the trivial zeta zeros ) and/or s =12 + i ( location of the non-trivial zeta zeros). In this case the RH would betrue and the non-trivial Riemann zeta zeros are given by sn = 12 + in and1sn = sn = 12 in . We are going to prove next why Case A does and cannotoccur, therefore the RH is true because we are left with case B.

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The inner product are dened as follows,

f |g = 0

f gdtt

.

Based on this denition the inner product of two eigenfunctions of D1 is

s 1 |s 2 =

0

e2V ts 12 +2 k1dt

=2l

Z 2l

(2k s12 ) ,. (1.16)

where we have denoted s12 = s1 + s2 = x1 + x2 + i(y2 y1) and used the expres-sions for the Gauss-Jacobi theta function and the denition of the completedzeta function Z [s] resulting from the Mellin transform as shown below.

We notice thats 1 |s 2 = s o |s , . (1.17)

thus, the inner product of s 1 and s 2 is equivalent to the inner product of s oand s , where so = 1 / 2 + i0 and s = s12 1/ 2. The integral is evaluated byintroducing a change of variables t l = x (which gives dt/t = (1 /l )dx/x ) andusing the result provided by the Gauss-Jacobi Theta given in Karatsuba andVoronins book [2]. The completed function Z [s] in eq-(1.16) can be expressedin terms of the Jacobi theta series, (x) dened by eqs-(1.5, 1.6) as

0

n =1

en 2 x xs/ 21dx =

0 xs/ 21(x)dx=

1s(s 1)

+ 1 [xs/ 21 + x(1s ) / 21] (x)dx= Z (s) = Z (1 s),

.

(1.18)where the completed zeta function is

Z (s) s/ 2 (s2

) (s), . (1.19)

which obeys the functional relation Z (s) = Z (1 s), which is a self-dualityrelation [26].In [10] we recurred to an infinite family of H A , H B operators associated

with an infinite family of potentials V jm (t) corresponding to an innite familyof theta series with the advantage that no regularization of the inner products is

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necessary. Another salient feature is that the pseudo-norm < jms | CT |jms >is not null ( see below eq-(1.28) ) as result that the zeta function (s) hasno zeros at s = 12 2m, m = 1 , 2, 3, ...., . The relevance of the behavior of ( 12 2m) = 0 , m = 1 , 2, 3, ..., is that it automatically avoids looking at thebehavior of zeta at s = 1 / 2. Armitage [3] has found a zeta function L (s) denedover the algebraic number eld L that has a zero at s = 1 / 2 and presumablysatises the RH . This nding would not be compatible with the result of eq-(1.16) and which was based on a regularized inner product. Therefore, the welldened inner product where no regularization is needed leads to the result (seebelow eq-(1.28) ) < jms | CT |jms > ( 12 2m) = 0 , m = 1 , 2, 3, ..., and which is no longer in variance with the behaviour of the zeta function L (s)dened over the algebraic number eld L and that has a zero at s = 1 / 2 [3].

Analogous results follow if we had dened a new family of potentials V 2j (t)in terms of a weighted theta series 2j (t) and whose Mellin transform yieldsthe innite family of extended zeta functions of Keating [4] and their associatedcompleted zeta functions as shown by Coffey [5]. The Hermite polynomialsweighted theta series associated to 2 j = even degree polynomials are dened by

e2V 2 j ( t ) = 2j (t) n =

n = (8)j H 2j (n2t ) en 2 t . (1.20)

and are related to the potentials V 2j (t) which appear in the denitions of thedifferential operators (1.1, 1.2). The weighted theta series obeys the relation

(1)jt 2j (1t

) = 2j (t). (1.21)

Only when j = even in (1.21) one can implement CT invariance in the newfamily of Hamiltonians H A , H B associated with the potentials V 2j (t) of (1.20)because H A s (t) = s(1 s)( t) and H B s ( 1t ) = s(1 s) s ( 1t ) would only bevalid when j = even as a result of the relations (1.1, 1.2, 1.3) and (1.20, 1.21).

The Mellin transform based on the weighted 2j (t) [5] requires once again toextract the zero mode n = 0 contribution of 2j (t) (to regularize the divergentintegrals) in order to arrive at

012

[ 2j (t) (8)2j H 2j (0) ] t s/ 21 dt = P j (s) s/ 2 (s2

) (s), Re s > 0.

(1.22)in the denition of the (regularized) inner products of the eigenstates associ-

ated to the new potentials (1.20). The polynomial pre-factor in front of thecompleted Riemann zeta Z (s) = s/ 2 ( s2 ) (s) is given in terms of a termi-nating Hypergeometric series [5]

P j (s) = (8 )j (1)j(2 j )! j ! 2

F 1( j,s2

;12

; 2). (1.23)

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The orthogonal states s n (t) to the ground state s o (t) ( so = 12 + i0) will nowbe enlarged to include the nontrivial zeta zeros and the zeros of the polynomialP j (s).

The polynomial P j (s) has simple zeros on the critical line Re s = 12 , obeysthe functional relation P j (s) = ( 1)j P j (1 s) and in particular P j (s = 12 ) = 0when j = odd [5]. It is only when j = even that P j (s = 12 ) = 0 and when wecan implement CT invariance resulting from the relation (1.21) and which isconsistent with the results of eqs-(1.8, 1.10a. 1.10b).

In order to avoid the regularization of the integrals involving the Mellintransform (1.22), we proposed another family of theta series where no regular-ization is needed in the construction of the inner products. There is a two-parameter family of theta series 2j, 2m (t) that yield well dened inner productswithout the need to extract the zero mode n = 0 divergent contribution. Given

e2V 2 j, 2 m ( t ) = 2j, 2m (t) n =

n = n2m H 2j (n2t ) en 2 t . (1.24)

when m = 0, the zero mode n = 0 does not contribute to the sum and theMellin transform of 2j, 2m (t) , after exploiting the symmetry of the even-degreeHermite polynomials, is [4], [5]

0 [ 2n =

n =1n2m H 2j (n2t ) en 2 t ] ts/ 21 dt =

2 (8)j P j (s) s/ 2 (s2

) (s 2m); Re s > 1 + 2 m, m = 1 , 2,.... (1.25)In order to nd the analytical continuation of the Mellin transform (1.25) forall values of s in the complex plane we must use the analytical continuation of (s) was found by Remann in his celebrated paper. A Poisson re-summationformula for 2j, 2m (t) (1.24) leads to similar modular behaviour as eq-(1.21)and only when j = even one can implement CT invariance in the new family of Hamiltonians H A , H B associated with the new potentials V 2j, 2m (t) of (1.24)

Therefore one has now at our disposal a well dened inner product of thestates s (t) (without the need to regularize it by extracting out the zero n = 0mode of the theta series). In particular the inner product of the states s (t)with the shifted ground states 1

2 +2 m (t), m = 1 , 2,.... corresponding to thepotentials in (1.24), by recurring to the result (1.25) and following similar stepsas in (1.16) is

< 12 +2 m (t) | s (t) > = 2 (8) j P j (s + 2 m) ( s +2 m )/ 2 ( s + 2 m2 ) (s).

(1.26)this result requires fixing uniquely the values l = 2; k = 14 . The nontrivialzeta zeros sn correspond to the states s n (t) orthogonal to the shifted groundstates 1

2 +2 m (t) in eq-(1.26) :

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< 12 +2 m (t)

|s n (t) > =

2 (8) j P j (sn + 2 m) ( s n +2 m )/ 2 (sn + 2 m

2) (sn ) = 0; m = 1 , 2, 3, .....

(1.27)It remains to prove when l = 2, k = 14 and s12 = s1 + s2 = s1 +(1 s1 ) = 1that

< s | CT |s > = < s || 1s > =

0 [ 2n =

n =1n2m H 2j (n2t ) en 2 t ] t 2( s 12 +2 k )2 l 1 dt =

2 (8) j P j (s =12

) 1/ 4 ( 14

) (12 2m) = 0; j = even, m = 1 , 2, 3, .....

(1.28)Hence, one arrives at a denite solid conclusion based on a well dened innerproduct : because ( 12 2m) = 0 when m = 1 , 2, .... , and P j ( 12 ) = 0 when j = even in eq-(1.28), the pseudo-norm < s | CT |s > = 0, and thisrules out case A in eq-(1.14) , and singles out case B in eq-(1.15) leading tothe conclusion that E s = s(1 s) = real s = 12 + i ( and/or s = real ),and consequently the RH follows if, and only if, CT invariance holds. The keyreason why the Riemann hypothesis follows is due to the CT invariance of theHamiltonians H A , H B and that the pseudo-norm < s |CT|s > is not null .Had the pseudo-norm < s |CT|s > been null , the RH would have been false.It remains to be seen whether our procedure is valid to prove the grand-RiemannHypothesis associated to the L-functions.

2 The Dirac and Schroedinger Operators thatreproduce the zeta zeroes

The previous section was devoted to a family of scaling operators needed in theconstruction of a pair of non-Hermitian Hamiltonians, involving ( t) functions,whose spectrum E s = s(1 s) = E s = s(1 s) was shown to be real valuedresulting from CT invariance, and whose solutions for s are s = 12 + i and/ors = real , showing how the Riemann Hypothesis is a physical realization of CT invariant QM. In this section we will nd the Dirac-like Operator (with a poten-tial V (x)) in one-dimension whose spectrum reproduces exactly the imaginaryparts (ordinates) n = E n of the zeta zeroes : ( 12 iE n ) = 0. At the endwe will also provide a different potential V (x) associated with a Schroedingeroperator in one-dimension that provides the same spectrum n = E n

The Dirac-like equation in one-dimension in the presence of a potential V (x)is

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{ i

x+ V (x)

}E (x) = E E (x); h = c = 1 . (2.1)

where the one-dim Clifford algebra with 2 1 = 2 generators is realized in termsof the unit element 1 and a 1 1 matrix whose entry is i. Eq-(2.1) is theoperator representation of the (constraint) dispersion relation

P (x) + V (x) = E, when P i

x(2.2)

such that the Bohr-Sommereld quantization condition yields the number of energy levels E 1 , E 2 ,...,E n (the number of the rst n zeta zeroes on the criticalline)

2

x n

x o =0 P (x) dx =2

x n

x o =0[ E n V (x) ] dx =

2

E n

V o(E n V ) dxdV dV = N (E n ) N (V o). (2.3)

We have set the lower integration limits at x = 0 because we assume thatthe potential is symmmetric. In fact, the potential will also turn out to bemultivalued . The potential used by Wu-Sprung [13] for the Schroedinger oper-ator

{ h2

2m 2

x 2+ V W S (x) } = E ; h2 = 2 m = 1 . (2.4)

turned out to be symmetric V W S (x) = V W S (x) for the choice of the averageenergy level counting function given by

N (E ) =E 2 [ log(

E 2 ) 1 ] +

78 . (2.5)

(where log is the natural Neper logarithm in the Euler number base) afterrecurring to the solutions to Abels integral equation of the rst kind obtainedafter differentiation w.r.t the E parameter of the Bohr-Sommereld quantizationcondition

2

E

0E V dx =

2

E

V o

E V dxdV

dV = N (E ) N (V o). (2.6)where V o = V (x = 0) was chosen to obey the boundary condition N (V o) = 0

N (V o) = V o2 [ log( V o

2 ) 1 ] + 78 = 0 V o3.10073 . (2.7)A differentiation of eq-(2.6) w.r.t to E (using Liebnitz rule) gives

2

ddE

E

V o

E V dxdV

dV =1

E

V o

1E V

dxdV

dV =

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dN (E )dE

=1

2log(

E 2

). (2.8)

The above equation belongs to the family of Abels integral equations associatedwith the unknown function f (V ) (dx/dV )

J [dxdV

] =1

() E

0

(dx/dV )(E V )1

dV =1

2log(

E 2

); 0 < < 1 (2.9)

The reason one had to differentiate eq-(2.6) w.r.t E is to enforce the condition0 < < 1 . Abels integral equation is basically the action of a fractionalderivative operator J [12] , for the particular value = 12 , on the unknownfunction f (V ) = ( dx/dV ) . Inverting the action of the fractional derivativeoperator (fractional anti-derivative) yields the solution for

dxdV

=1

(1 )d

dV V

0

12

log (E 2

)1

(V E )dE. (2.10a)

After setting the value = 12 in (2.10a) , Wu-Sprung [13] found the solutionin terms of quadratures for the dx/dV function, and a subsequent integrationw.r.t V , gives

x = x(V ) =1 V V o log (

V o2e2

) +1

V log[ V + V V oV V V o]. (2.10b)

where e = 2 .71828... is Eulers number and when V = V o xo = x(V o) =0 consistent with the condition V (x = 0) = V o 3.10073 . The sought

after potential V W S (x) that reproduces the average level density of zeta zeroes(energy eigenvalues) given by N (E ) is implicitly given from the relation x(V )upon inverting the function. It was after the tting process of the rst 500Riemann zeta zeroes on the critical line when Wu-Sprung found numericallythat a fractal shaped potential (obtained as a perturbation of the smoothV W S (x)) of dimension d = 1 .5 was needed. A further tting of the rst 4000zeroes furnished identical results for the fractal dimension d = 1 .5 [15] associatedwith the shape of the potential.

Based on these ndings we proposed within the context of SupersymmetricQM a Weierstrass fractal function [16] as the fractal-shaped corrections to thesmooth potential (2.10b) and consistent with the numerical ndings by [13],[15] in order to model the fractal behaviour of the potential that tted thosezeta zeroes. Later on, Slater [17] performed an exhaustive detailed numerical

analysis of our Weierstrass fractal function (and other fractal functions) to nda numerical t for the rst n = 25 , 50, 75,.... zeta zeroes.The relevant feature of the expression for x(V ) (2.10b) is that it is explicitly

given in terms of square roots (quadratures), such that changing the signs of thesquare roots containing the variable V will yield a change of sign : x(V ) =x(V ) consistent with the assumption that V was symmetric V (x) = V (x).

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One can verify why this must be so from Abels solution (2/10a) . If ( dx/dV )changes to

(dx/dV ) by replacing x

x, leaving V xed, one must take

the minus sign of the 1/ V E terms appearing in the r.h.s of (2.10a) when = 12 .In general, when one replaces the average level counting function N (E ) by

the more general expression N (E ) obtained by Riemann-von Mangoldt formula(given below in eq-(2.12)) one has the solution ( for = 12 )dxdV

=1

(1 )d

dV V

0

d N (E )dE

1(V E )

dE

x(V ) x(V o) =1

(1 ) V

0

d N (E )dE

1(V E )

dE ; =12

. (2.11)

At the end of this section we shall return to the solution (2.11) corresponding

to the Schroedinger operator. Solutions to eq-(2.11) for a truncated version of the Riemann-von Mangoldt formula (2.12), where the oscillatory terms and theintegral terms were dropped, was given by Slater [17] using the MathematicaIntegrator package.

By recurring to a Dirac-like operator it allows to use the full expressionfor number of zeroes (energy levels) N (E ), including the uctuating/oscillatoryterms, leading to a symmetric and multi-valued potential due to the oscillatoryterms in N (E ). The coordinate function x(V ) is assumed to be single-valued,but its inverse, the potential V (x) is not necessarily single-valued, and in fact,it will turn out to be multi-valued. The typical example is the sine functionx = sin (V ) (single-valued) whose inverse V = arcsin (x) is multi-valued.

A knowledge of the functional form of the number of zeroes N (E ) in theabove integral-differential equation (2.3) gives the potential V (x) implicitly.Let us write the functional form for N (E ) to be given by the Riemann-vonMangoldt formula which is valid for E 1

N RvM (E ) =E 2

[ log(E 2

) 1 ] +78

+1

arg [ (12

+ iE ) ] +1

(E ). (2.12)

where the (innitely many times) strongly oscillating function is given by theargument of the zeta function evaluated in the critical line

S (E ) =1

arg [ (12

+ iE ) ] = lim 01 I m log [ (

12

+ iE + ) ]. (2.13)

the argument of (12 + iE ) is obtained by the continuous extension of arg (s)along the broken line starting at the point s = 2 + i 0 and then going to the

point s = 2 + iE and then to s = 12 + iE . If E coincides with the imaginarypart of a zeta zero, then

S (E n ) = lim 012

[S (E n + ) + S (E n )]. (2.14)

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An extensive analysis of the behaviour of S (E ) can be found in [20]. In par-ticular the property that S (E ) is a piecewise smooth function with discontinuitiesat the ordinates E n of the complex zeroes of (sn = 12 + iE n ) = 0. When E passes through a point of discontinuity, E n , the function S (E ) makes a jumpequal to the sum of multiplicities of the zeta zeroes at that point. The zeroesfound so far in the critical line are simple [22]. In every interval of continuity(E, E ), where E n < E < E < E n +1 , S (E ) is monotonically decreasing withderivatives given by

S (E ) = 1

2log (

E 2

) + O(E 2); S (E ) = 1

2E + O(E 3). (2.15)

The most salient feature of these properties is that the derivative S (E ) blowsup at the location of the zeta zeroes E n due the discontinuity (jump) of S (E ) atE n . Also, the strongly oscillatory behaviour of S (E ) forces the potential V (x)to be a multi-valued function of x.

The expression for (E ) is [20]

(E ) =E 4

log (1+1

4E 2) +

14

arctan (1

2E )

E 2 0

(u) du(u + 1 / 4)2 + ( E/ 2)2

.

(2.16)with (u) = 12 {u}, where {u}is the fractional part of u and which can bewritten as u [u], where [u] is the integer part of u. In this way one can performthe integral involving [ u] in the numerator by partitioning the [0 , ] interval inintervals of unit length : [0 , 1], [1, 2], [2, 3], ..... [n, n + 1] ,... . The denite integralwhen the upper limit is bound by an ultra-violet regulator is

E 2

0

(u) du(u + 1 / 4)2 + ( E/ 2)2

=E 4

log [(E/ 2)2 + ( + 1 / 4)2

(E/ 2)2 + (1 / 4)2]

[]

n =1n [ arctan (

4n + 52E

) arctan (4n + 1

2E ) ]

34

[ arctan (4 + 1

2E ) arctan (

12E

) ] (2.17a)

It is the Euler-Maclaurin summation formula

N 1

k =1

f k = N

0f (k) dk

12

[f (0) + f (N )] +

112

[f (N ) f (0)] 1

720[f (N ) f (0)] + ..... (2.17b)

that permits the exact evaluation of the limit of the expression (2.17a):the divergent terms E log() in (2.17a) cancel out exactly leading to the (E )terms of (2.16)

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(E ) = E

4 +

E 2

8 [ arctan (

5

2E ) arctan (1

2E ) ] +132

[ 33 arctan (1

2E ) 25 arctan (

52E

) ] +E 16

[ 5 log (1+25

4E 2) log (1+

14E 2

) ] +

112

[ arctan (5

2E ) arctan (

12E

) ] + ........ (2.17c).

For large E , a Taylor expansion of (E ) gives

(E ) =148

1E

+ O(1

E 3). (2.17d)

Thus the leading term of (E ) is of the order (1 /E ) as expected. However, itis important to keep all the terms involving E given by (2.17c) when E is notlarge.

We must search now for solutions to the integral equation associated withthe Dirac-like operator in one-dimension

1

E

V o( E V )

dxdV

dV = N (E ) N (V o) (2.18)associated with the unknown function f = f (V ) dxdV and subject to theboundary condition V (x = 0) = V o . i.e. the integral transform of f (V ) denedby eq-(2.18) is the counting function N (E ). The solutions to Abels integralequations will not be necessary in our case to nd dx/dV . What should thechoice of V o be ? To answer this question we need to discuss the followingpoints. The integral (2.18) is trivially zero when the upper limit E coincideswith V o , which is consistent with the trivial fact : N (E = V o) N (V o) = 0.

Despite that the Riemann-van Mangoldt expression (2.12) is only valid forE 1, one can still verify by inspection that when V o = 0 N (E = V o =0) = 0. This can be seen if one chooses the argument of (1/ 2) = 1.46 tobe given by , instead of . With this choice for the argument and takingarctan () = 2 , then (2.12) becomes

N (E = 0) =78

+1

() +1

42

= 0 . (2.19)

Had one chosen the argument one would have N (E = 0) = 2 which is thewrong answer since there are no zeros at (1/ 2). The choice V o = 0 is a verynatural one from the physical point of view and compatible with the E = 0ground state of Supersymmetric Quantum Mechanics (SQM) in one-dimension.The super-potential W (x) in SQM vanishes at x = 0 if Supersymmetry is notbroken.

Upon taking derivatives on both sides of eq- (2.18 ) w.r.t to E gives

2

E

0

dx(V )dV

dV =2

x

0dx =

2x(E )

=d N (E )

dE (2.20)

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Notice that despite the derivatives N (E ) blow up at the location of the zetazeroes E = E n , due the discontinuity (jump) of S (E ) at E n , the expression(2.20) is nevertheless correct because it just means that the function x(E ) alsoblows up x(E n ) = when E = E n . Therefore, the fact that N (E ) blows up ata discrete number of locations E = E n does not preclude us from differentiatingboth sides of eq-(2.18) w.r.t to E .

From the above relation (2.20) we will show that the solution to the integralequation (2.18) is

2x

=2x(V )

=

d N (V )dV (V ). (2.21)

where N (V ) has the same functional form as N (E ). The physical interpre-tation of (2.21) is that the coordinate function x = x(V ) is just proportionalto the density of zeroes (V ) (times / 2) : the number of zeros per unit of energy. On dimensional grounds this makes sense, since length has the dimen-sions of inverse of energy when h = c = 1. Therefore, one can infer that whenV = E n x = x(E n ) = xn = for all values of n = 1 , 2, 3, ..... due to thesingular behaviour of the derivatives N (V = E n ) at the ordinates of the zetazeroes resulting from the discontinuity of the argument of the zeta function atE n . The last expression (2.21) for x(V ) furnishes the sought-after potentialV = V (x) in implicit form. x(V ) is single-valued but V (x) is multi-valued. Inparticular, V (x = ) = E n , n = 1 , 2, 3, ..... .Equipped with the known expression for the functional form of N (V ) (2.12)(after replacing E for V ) the quantization condition (2.18) reads

E

V o(E V )

d2 N (V )dV 2

dV = N (E ) N (V o) . (2.22)Taking derivatives on both sides of (2.22) w.r.t to E and using the most generalLiebnitz formula for differentiation of a denite integral when the upper b(E )and lower b(E ) limits are functions of a parameter E :

ddE

b(E )

a (E )f (V ; E ) dV =

b(E )

a (E )(

f (V ; E )E

) dV +

f (V = b(E ) ; E ) (d b(E )

dE ) f (V = a(E ) ; E ) (

d a(E )dE

). (2.23)

leads to

E

V o

d2 N (V )dV 2

dV =d N (E )

dE . (2.24)

since the lower limit V o is taken to be independent of E and the integrandvanishes in the upper limit V = b(E ) = E . The integral (2.24) is straightforward

d N (V )dV

(V = E ) d N (V )

dV (V o) =

d N (E )dE

. (2.25)

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from which one infers that one must have N (V o) = d N (V )dV (V o) = 0 since thefunctional forms of N

(V ) and

N (E ) are the same .

However, there is a potential problem because there is no assurance thatthe function N (V ) obeys the condition N (V o) = 0 unless one chooses the valueof V o to be the solution to the transcendental equation

12

log (V o2

) +1 I m

i ( 12 + iV o) ( 12 + iV o)

+1

4log (1 +

14V 2o

) 3

21

1 + 4 V 2o 1

2 0 (u) du(u + 1 / 4)2 + ( V o/ 2)2 + V o2

0

(u) (V o/ 2) du[ (u + 1 / 4)2 + ( V o / 2)2 ]2

= 0 .

(2.26)If, and only if, there is a real -valued solution V o to eq-(2.26) that xes thevalue of the zero-point energy V o and that falls in the range 1

V o < E 1 , then

one has that x(V ) = 2 (d N (V )/dV ) yields the potential V (x) in implicit formreproducing the ordinates of the zeta zeros for the spectrum E n .However, if there is no real-valued solution V o to the transcendental equation

(2.26) that falls in the range 1 V o < E 1 , then one can go ahead and truncatethe upper limit of the denite integral appearing in the denition of (E ) in eqs-(2.16, 2.17) by introducing an E -dependent ultraviolet cutoff = ( E ), suchthat the zero derivative condition is modied from the form given by (2.26) tothe one given by

N (V o , (V o)) =1

2log (

V o2

) +1 I m

i ( 12 + iV o) ( 12 + iV o)

+1

4log (1 +

14V 2o

) 3

21

1 + 4 V 2o 1

2 ( V o )

0

(u) du(u + 1 / 4)2 + ( V o/ 2)2

+V o2

( V o )

0

(u) (V o/ 2) du[ (u + 1 / 4)2 + ( V o/ 2)2 ]2

V o2

((V o))[ ((V o)) + (1 / 4) ]2 + ( V o / 2)2

(d(V )

dV )(V = V o) = 0 . (2.27)

Therefore, the above condition N (V o , (V o)) = 0 provides the necessary con-straint between V o and ( V o) to satisfy our goal. It is customary in the Renor-malization Group process in Quantum Field Theories (QFT) to introduce anenergy cut-off; here ( E ) is a running and increasing function of E which tendsto innity when E .To sum up, if there is a real -valued solution V o to eq-(2.26) that xes thevalue of the zero-point energy V o and that falls in the range 1

V o < E 1 , then

x(V ) = 2 (d N (V )/dV ) yields the potential V (x) in implicit form. If there is noreal-valued solution V o to the transcendental equation (2.26) that falls in therange 1 V o < E 1 , then one truncates the upper limit of the denite integralleading to a modied equation (2.27) and x(V ) = 2

d N (V, ( V ))dV determines thepotential implicitly in terms of the cut-off function ( V ).

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Next we describe how one determines the functional form of the cut-off function ( V ) in such a case. Because ( E ) is a cutoff-function that runs withenergy E , one has now enough freedom to impose the exact conditions

N (E n ; (E n )) N (V o ; (V o)) = n; n = 1 , 2, 3, ..... ; 1 V o < E 1(2.28a)when E 1 , E 2 , E 3 , .......E n , E n +1 , ..... are the (positive) imaginary parts (ordi-nates) of the zeta zeroes in the critical line. In order to evaluate N (V ) atV = E n due to the discontinuity of the uctuating term S (E ) of (2.12) at E none must take the arithmetic mean as described by eq-(2.14). The upper limit of the values of V o should be bounded by the rst zero E 1 avoiding having potentialsingularities in eq-( 2.27) due to the zeros of zeta appearing in the denominatorof second term. The lower bound of V o should be 1 since the domain of validityof the Riemann-van Mangoldt expression is E

1.

If one were to replace the values ( E n ) = n for = one may rewriteeq-(2.28) as follows

N (E n ; n ) N (V o ; (V o)) = [ N (E n ; = ) + ( n , E n ) ] N (V o ; (V o)) =(n n ) + ( n , E n ) N (V o ; (V o)) = n. (2.28b)

where the number of levels (zeros) just below the n-th zero given by the Riemann-van Mongoldt expression are N (E n ; = ) = n n (if the Riemann Hypoth-esis is true), and n is a fraction such that 0 < n < 1. The positive-denitequantity ( n , E n ) is the decit value of the integral appearing in eqs-(2.16,2.17) given by ( E n / 2) n (.... ).Finally, one can derive implicitly the potential of the Dirac-like operatorthat reproduces the zeta zeros from

2x

=2x(V )

=

d N (V ; (V ))dV

= N (V ; (V ))

V +

d(V )dV

N (V ; (V )) (V )

xo = x(V o) =2

d N (V ; (V ))dV

(V = V o) = 0 .

Next we describe how to solve the system of eqs-(2.27-2.28). Firstly, onebegins by truncating the series expansion for the cut-off function ( V ) as follows

(V ) =k=

k=0

ak V k (V ; N ) =k= N

k=0

ak V k

(d(V, N )

dV ) =

k= N

k=0

ak k V k1 (2.29)

We are going to display two case scenarios how to solve eqs-(2.27-2.28). Inthe rst case we are going to drastically simplify these equations by choosing

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V o = 0, despite that the domain of values for V o in the denition of N (V o) givenby the Riemann-van Mongoldt formula should be 1

V o < E 1 . In the secondcase scenario we shall enforce the latter conditions on V o . Hence, eqs-(2.27-2.28)are automatically simplied by setting V o = 0 N (V o = 0 ; o) = 0, whichfollows from eq-(2.19 ), so now the value of o = ( V o = 0) is determined fromthe relation (2.27) for V o = 0

1

2log (4) +

1 I m

i ( 12 ) ( 12 )

32

12

o

0

(u) du(u + 1 / 4)2

= 0 . (2.30)

after noticing a cancellation between the singular log(V o = 0) log(V o = 0)terms associated with the rst and third terms of (2.27) resulting from therelation lim V o 0 (1/ 4) log(1+

14V 2o

) (1/ 2) log(V o = 0) = + . The valueof the integral in (2.30) can be inferred from (

2/E )

the value of the integral

in eq-(2.17). When E 0, it requires the use of LHopitals rule giving a niteresult for a nite value of o . The values ( 12 ) = 3.92265; ( 12 ) = 1.46 ineq-(2.30) yields a negative value for o given by

0.0787 =12

o

0

(u) du(u + 1 / 4)2

= 12

0

o

(u) du(u + 1 / 4)2

; with 14

< o < 0.

The quantization conditions (2.28a, 2.28b) corresponding to N (V o = 0; o) = 0,and ( V o = 0) o < 0 given by the solution to eq-(2.30), and by writing(E n ) n , become

N (E n ; n ) N (V o = 0 ; o) = N (E n ; n ) = n; n = 1 , 2, 3, ......... ( n , E n ) =

E n2 n

(u) du[ (u + 1 / 4)2 + ( E n / 2)2 ]

= n = n N (E n ; = ).(2.31)

( n , E n ) is positive denite because the contributions to the integral in allof the intervals [ [ u], [u] + 1 ] which do not contain n are all positive due tothe increasing values of the denominator, whereas the magnitude of the values(u) = 12 {u}are symmetrically distributed about the midpoint of the intervalswhile being positive and negative denite in the intervals [ [ u], [u] + 12 ], [ [u] +12 , [u] + 1 ] respectively. The integral in the interval [ [ n ], [n ] + 1 ] is positive,negative or zero depending on the location of n . One can always choose thevalue of n to obey the relations in eq-(2.31).

To sum up, given a set of N + 1 integers n = 0 , 1, 2, 3, ......N bounded by N ,

Eqs-(2.30, 2.31) yield a system of N + 1 equations which in principle determinethe values of the N + 1 unknown cut-off parameters o , 1 , 2 ,....... , N interms of the ordinates of the zeta zeros E 1 , E 2 , E 3 , .....,E N . The value of ohas already been xed from eq-(2.30). Eqs-(2.31) determine the remaining onesn , n = 1 , 2, 3, ....N .

Finally, after solving eqs-(2.30, 2.31), the dening relations

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(V o ; N ) =k= N

k =0

ak (V o)k = o (V o = 0 ; N ) = ao = o . (2.32)

and

(V = E n ; N ) =k = N

k=0

ak (E n )k = n . (2.33)

yield a linear system of algebraic equations for the N coefficients a1 , a 2 , .... , a N associated with the truncating series (2.29), and whose solutions are given interms of the imaginary parts of the zeta zeroes E n and the values of the cut-off parameters ( V o = 0) = o , (E n ) = n . The latter cut-off parametershave themselves been determined from the solutions to eqs-(2.27, 2.28). Thesolutions for the coefficients ak can be written compactly in terms of the vander Monde determinant of the ( N + 1) (N + 1) matrix comprised of N rowswhose entries in the n-th row are 1 , E n , E 2n , E 3n , ........,E N n , for n = 1 , 2, 3,....,N .The rst row has entries 1 , 0, 0, 0, ....., 0 since V o = E o = 0, so the total numberof rows and columns is N + 1. The van der Monde determinant is

= (E i E j ), for i > j. (2.34)The other determinants involved in the solutions k correspond to the ( N +1) (N + 1) matrices obtained by replacing the k-th column by a column comprisedof the entries o , 1 , 2 , 3 , ......., n . The solutions for the coefficients that denethe cut-off function ( E, N ) at level N are compactly written as

a (N )k = k ( o , {n }; V o = 0; {E n })(E i E j ). i > j, i, j = 0 , 1, 2, 3, .......N. (2.35)

with E o = V o = 0. If the large N limit converges

lim N a(N )k ( o , 1 , 2 ,...., N ; E 1 , E 2 , .....,E N ) ak , a fixed point.(2.36)

to a xed point, the full edged energy-dependent cut-off function ( E ) is de-termined by the innite series

(E ) =k=

k=0

ak E k . (2.37)

which is dened in terms of the innite number of coefficients given by theinnite number of fixed points ak . The spectral statistics of Random MatrixModels in the large N limit have been known for a long time to have deep con-nection to the zeta zeroes since Montgomery-Dyson found the pair-correlationfunctions of the ordinates of the zeta zeroes with normalized spacings in terms

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of an integrand involving the function 1 ( sinxx )2 . This function is the pair cor-relation function for the eigenvalues of very large Random Hermitian matricesmeasured with a Gaussian measure (the Gaussian Unitary Ensemble) [21].

Since xed points in Renormalization Group (RG) techniques in QFT areubiquitous, it is warranted to explore the connections among the putative xedpoints ak with the xed points associated with the beta function in QFT. ARG analysis was performed by Peterman [23] to shed some light as to why thedensity of primes numbers decreases as 1 /logx . The zeta function has also beenused extensively in Regularization methods (of innities) in QFT, see [24] andreferences therein. The Russian doll Renormalization group has been foundto have connections to the RH [19].

Finally, once the cut-off function ( V ) is constructed from the denition

(V ) =k =

k=0

ak ( o ,

{n

}; V o = 0;

{E n

}) V k . (2.38)

for all values n = 1 , 2, 3, ...., , the sought-after potential is implicitly deter-mined from the fundamental result2x

=2x(V )

=

d N (V ; (V ))dV

. (2.39)

where the functional form of N (V ; (V )) is given by the Riemann-von Mangoldtformula (2.12) by replacing E V and by inserting the energy-dependentcut-off (V ) found in eq-(2.38) into the upper limit of the integral appearingin the denition of the (E ) terms in eq-(2.16). Naturally, due to numericallimitations, the potential can only be constructed iteratively , level by level,N, N + 1 , N + 2 , ......, . Another salient feature to look for is to verify thatthe family of coefficients a

(N )k does indeed converge to the xed points values akwhen N . This is where the results of large N Random Matrices methodsare relevant.

The second case scenario is more complicated to solve if one forces V o to liein the domain 1 V o < E 1 . Choosing a particular value of V o = V o in thatrange, one has for eqs- (2.29)

(V o ; N ) =k= N

k =0

ak (V o )k = o . (2.40)

and

(V = E n ; N ) =k = N

k=0

ak (E n )k = n . (2.41)

when V o = V o = 0, one has new solutions for the coefficients ao , a 1 , a 2 , ......a N

ak = ak (o , n ; V o , E n ) = k ( o , {n }; V o , {E n })

(E i E j ); i > j (2.42)

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for i, j = 0 , 1, 2, 3, .......N and where the rst row of the matrix involved inthe van der Monde determinant now must include the 1 , V o , (V o )2 , ......., (V o )N

terms.Equipped with the above solutions ak (2.42) one inserts them into the ex-

pression for the derivative term

(d(V, N )

dV ) (V o ) =

k = N

k =0

ak ( o , n ; V o , E n ) k (V o )k1 . (2.43)

which appears in eq-(2.27). The latter equations (2.27, 2.40, 2.41, 2.42, 2.43)combined with eq-(2.28), where now N (V o , o ) = 0, furnishes a system of N +1equations which determines in principle the numerical values for the cut-off parameters o , n s . From these latter values one reconstructs the the akcoefficients from eqs-(2.42), and which in turn, will determine the sought-aftercut-off function ( V ; N ) eq-( 2.29) (at level N ), so that nally one can write thefull explict form of N (V, (V )) (given by the Riemann-van Mongoldt formula).Its derivative yields d N (V, ( V ))dV = (2 x(V )/ ) which nally furnishes the form of the sought-after potential V (x) (implicitly). Naturally, the equations to solvein this second case scenario are far more difficult that the ones when one simplychooses V o = 0 simplifying drastically these calculations.

There is a more general third case scenario when one has N + 2 undeter-mined parameters V o , o , n , n = 1 , 2, 3,....,N at each level N which areconstrained to obey N +2 equations given by the N +1 eqs-(2.27, 2.28), plus anadditional extra condition involving the second derivatives N (V o , o ) = 0.The procedure to solve this far more complicated system of N + 2 equations isstill similar to the second case scenario, the only difference is that now V o is

not put in by hand, but instead is another unknown parameter to be determinedfrom the solutions of these N +2 equations. The open question remains whetheror not the solutions for V o fall in the range 0 V o < E 1 . Even perhaps, thesolution for V o might be negative. Out of these three cases, the simplest oneto follow is the one when one takes V o = 0 which simplies drastically all thecalculations and allows us to provide with actual numerical results. The fullnumerical analysis of eqs-(2.27, 2.28, 2.29, ...) is beyond the scope of this work.It requires very sophisticated computations.

To complete this subsection, we need to discuss the nature of the solutions(x) to the one-dim Dirac-like equation (2.1) on the line [ , ] given by

E (x) = o,E exp [ i x

P (x ) dx ] =

o,E exp [ i x

( E V (x ) ) dx ]. (2.44)

where E is a real-valued continuous parameter and o,E is a constant ampli-tude.

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The operator i/x would be self-adjoint in the full line [ , ], or inthe compact interval [ xa , xb], if one could impose suitable vanishing boundaryconditions on the above E (x) solutions at and/or at xa , xb. However,the solutions E (x) are just proportional to a pure phase factor so the s arenonvanishing for all value of x, unless one constrains the amplitudes o,E to zerothat will render the solutions trivial . Therefore, since one cannot nd nontrivialsolutions E (x) obeying the boundary conditions E (x = ) = 0 and/orE (x) = 0 at xa , xb, the operator i/x is not self-adjoint in [ , ], or inthe compact interval [ xa , xb], for the space of solutions given by (2.44). If one hada second order operator D2 = h

2

2m ( 2 /x 2) like the Schroedinger operator, there

are no problems with nding self-adjoint extensions. For example, a free particleinside a box of length 2 L admits normalizable wave-functions n (x)sin (

nxL )

obeying the suitable boundary conditions n (x = L) = 0.For these reasons we conclude that the self-adjointness property is not re-quired to fulll our goals. We saw in section 1 how by working with a pairof non -Hermitian Hamiltonian operators was sufficient to show why it was the

CT symmetry which forced the energy spectrum to be real : E s = s(1 s) =E s = s(1 s), leading to the only possible solutions s = 12 + i and/ors = real , and consistent with the fact that the Riemann zeta function hastrivial zeroes on the negative even integers and nontrivial zeroes in the criticalline Re s = 12 . For the Dirac-like operator (2.1) all we need is to impose P T symmetry where this time by T reversal symmetry we do not mean inversiont (1/t )log(t) log(t), but the standard t t symmetry used in P T symmetric QM.

The momentum p = dx/dt is invariant under P T symmetry since x andt both reverse signs, this means that i i under P T symmetry so that themomentum operator remains invariant p = ih(/x ). There is nothing strangeby having i change sign under P T symmetry since Clifford algebras in D = 1have two generators, the identity element 1 and the 1 1 matrix whose entryis just i, so that {, }= 2 i2 = 21 , if one takes the metric of the one-dimspace to be g11 = 1. Therefore under P T symmetry which impliesthat i i.This leaves us with having to impose the condition V (x) = V (x) on thepotential in order to have a P T symmetric Dirac-like operator i/x + V (x).In order to dene V (x) in the regions x < 0 one must choose the minus sign infront x(V ) = 2 d N (V, ( V ))dV . The positive sign selects the solutions in the regionx > 0. Furthermore, it is important to emphasize that the one-dimensionalJacobian (from the change of variables) is ( dx/dV ) in the x > 0 region, but it is

(dx/dV ) in the mirror x < 0 region. There is a crucial sign change to ensurethat the portion of the line-integral along the left region does not trivially cancelout the portion of the line-integral in the right region. The Bohr-Sommerfeldquantization rule involves the closed contour in phase space that in the case of a symmetric potential gives pdx = 4 o pdx = 2 n , thus care must be takenwith the signs of dx/dV .

To sum up this discussion : the self-adjointness (Hermitian) property is not

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required to prove the Riemann Hypothesis. What matters was the CT symmetryin section 1 and P T symmetry in this section related to the spectrum of theDirac-like operator in one-dimension. To nalize, once we extend the domain of V (x) to the region x < 0 by taking the mirror image of the potential constructedin this section; the solutions associated with the discrete family of zeta zeroesE n (embedded in the continuum of solutions) are is simply obtained by insertingthe value of E = E n inside the integrand of (2.45)

E n (x) =1

L exp [ i x

( E n V (x ) ) dx ]. (2.45)

where in the region x < 0 one must use the branch of the potential solutiongiven by x(V ) = 2 d N (V, ( V ))dV to ensure that indeed we are selecting solutionswhich obey V (x) = V (x). L is an infrared cutoff that is required so that thewave-functions E n (x) are square integrable on the line

lim L { L/ 2

L/ 2E n (x) E n (x) dx } = 1 . (2.46)

One must take the L limit after the integration (2.47) is performed andnot before otherwise one gets the trivial result for the wavefunctions = 0.Finally, when one evaluates the discrete family of wave functions at the cusps

points xn = + (where the values of the potential are V (x = xn = + ) = E n )one arrives at

E n (x = xn = ) =1

L exp [ i x n =

( E n V (x ) ) dx ] =

1L exp [ 2 i xn =

0( E n V (x ) ) dx ] =

1L exp {i [ N (E n ; (E n )) N (V o ; (V o)) ] } =

1L exp [ i n ] =

(1)nL . (2.47)as a direct result of the conditions in eq-(2.28) and eq-(2.3). Therefore, at thecusps points x = xn = + the wave functions E n (xn ) alternate in sign.This changing of sign is related to the presence of Gram points in the RiemannSiegel formula, with the only difference that the phase factors in (2.47) involvethe full-edged zeroes (discrete energy levels E n ) counting function N (E, (E ))whereas in the Riemann-Siegel formula only the average energy level countingfunction is used given by the rst two terms of eq-(2.12) [2].

The values of the wave-functions at the x = are simply E n (x = ) =1L . For even n the wavefunctions are periodic (with an infinite period) inthe sense that E n (x = ) = E n (x = + ). For odd values of n thewavefunctions are anti-periodic in the sense E n (x = ) = E n (x = + ).

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Therefore, the imaginary parts of the zeta zeroes E n in the critical line arethe only values among the E -continuum of values which obey the boundaryconditions E (x = ) = E (x = + ). This physical interpretationof the discrete values E n among the E -continuum must have bearing on theperiodic orbits associated with the chaotic Riemann dynamics whose periodsare multiples of logarithms of prime numbers as described by Berry and Keating,and based on their classical Hamiltonian H = xp [14]

Notice that if one were to replace the values n and ( V o) for = , onewould have for phases in the wave functions (2.47) the following values { N (E n ; = ) N (V o ; = ) } =

{n n N (V o; = ) } = n n o . (2.48)where now the zero-point energy V o is the one determined from the solutionto the transcendental equation (2.26). Therefore, if no cut-offs are set in theupper limits of the integral dening the (E ) terms (2.16, 2.17) there would bea nontrivial phase shift in the wave functions as shown in eq-(2.48) and onewould no longer have the nice periodicity (anti-periodicity) behaviour as before;i.e. one would have now a quasi -periodicity behaviour.

To nalize this section we return to the solution of Abels integral equation(2.11) ( = 12 ) in the Schroedinger operator case

x(V ) x(V o) =1

(1/ 2) V

0

d N (E )dE

1

(V E )dE. (2.49)

To control the divergences in the integral once again we may introduce a cut-off function ( E ) in the counting function and have d N (E, ( E ))dE inserted into theabove integral where the cut-off function ( E ) is dened by the series expansionof eq-(2.38). The coefficients ak are the fixed points of the large N limit of the family a (N )k (o , V o = 0; {n }, {E n }) given explicitly by the relations (2.35)involving the van der Monde determinant.

The values of the n s as functions of E n are obtained by solving eqs-(2.28),however, now the value of o is no longer determined from eq-(2.27), becausethat equation no longer applies, but it is left out as a free parameter thatis related to the integration constant x(V o = 0) = xo in the l.h.s of (2.48).Therefore, one has now, at each level N, N + 1 , N + 2 ..... a well behaved cutoff function ( E, N ) in the expression N (E, (E )) given by eq-(2.12) that can beinserted into the above integral (2.48), and provide solutions for x(V ) xo (x(V o = 0) = xo ), and which denes implicitly , the potential V (x) of the self-adjoint Schroedinger operator dened in the whole real line that reproduces the

zeta zeros.

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3 Area Quantization in Phase Space, Duality,

Spacetime Singularities, Renormalization Groupand Distribution of Primes and Zeta Zeros

To nalize this work , we will derive the Area Quantization condition in PhaseSpace An = n of the intervals [0 , E n ] for n = 1 , 2, 3, ...... and show whyarea quantization is one physical reason why the average distribution of primesdensity for very large x given by O( 1log x ), has a one-to-one correspondencewith the inverse average density of zeta zeros in the critical line. As the numberdensity of primes decreases asymptotically with large x as (1/log x ), the averagedensity of zeta zeros in the critical line increases asymptotically (for very largeE ) as 12 log(

E2 ). This nding is consistent with the results of Petermann [23]

who found the 1 /logx behaviour to be connected to the Renormalization Groupprogram in QFT.

In the previous section we found that the potential function V (x) obtainedimplicitly from eq-(2.40) turns out to be a multi-valued function of x whichrequires splitting the energy regions into different bands, branches, like a non-periodic crystal lattice

[0, E 1], [E 1 , E 2], [E 2 , E 3], ............ , [E n 1 , E n ], ...... (3.1)such that at the boundaries of those bands : x(V = E n ) = 2 N (V = E n ) = due to the discontinuity of the S (E ) term (2.12) at E n .

The left and right derivatives of x(V ) at V = E n are (dx/dV ) = which is also consistent with taking the second derivatives of the Heaviside stepfunction (E

E n ) = (E

E n ), since the counting function is dened by

N (E N ) = N 1 (E E n ). In the innitesimal region V = E n n , for asuitable innitesimal n (E n ) > 0, one expects a sudden jump of the function N (V, (V )), from values less than n, to values greater than n, while reachingthe precise value of n at V = E n due to the conditions imposed in (2.28). Thissudden jump is provided for by the S (E ) term in the Riemann-von Mangoldtformula.

On a separate problem, we exploited this singular behaviour of the deriva-tives of the Heaviside step function to construct a different solution to the staticspherically symmetric gravitational eld produced by a point mass M at r = 0than the standard text book solution. The solutions for the metric [32] werecontinuous except at the location r = 0 of the point mass, leading to a deltafunction for the scalar curvature R= (2 GM (r )/r 2) instead of R= 0. TheEuclideanized Einstein-Hilbert action coincided precisely with the Black HoleEntropy where the area of the horizon which has now been displaced at thelocation r = 0 + (due to the discontinuity of the metric at r = 0 ) is the usualvalue 4(2GM )2 . The area-radial function chosen was (r ) = r + 2 G|M |(r )so that (r = 0) = 0; (r = 0 + ) = 2 G|M |; (r = 0 ) = 2G|M | due to thedenition of the Heaviside step function : it is 1 for r > 0; 1 for r < 0, and

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is 0 for r = 0. This discontinuity has the same form as the discontinuity of theargument of ( 12 + iE ). For this reason we believe that John Nashs approachto the RH based on spacetime singularities was on the right path.

Between two consecutive cusps where the coordinate function blows upx(V ) = 2 N (V ; (V )) = at V = E n , E n +1 , lies a valley region where thereare inflection points of N (V, (V )) at the locations E

(n )

, within the intervalsE n < E

(n )

< E n +1 , such that N (E (n )

, (E (n )

)) = 0; i.e. ( dx/dV ) = 0 at thebottom of the valleys V = E (n )

, while (dx/dV ) = at the cusps E n . Onecan visualize the coordinate graph function x(V ) as an innitely long suspen-

sion bridge (from V = 0 to V = ) with innitely high poles/spikes x(V ) = at the specic locations V = E 1 , E 2 , E 3 , ....... , and with the suspension cablesfalling into the U -shaped valley regions in between.

With this picture in mind, the areas An in the Phase Space comprised by thisnon

periodic crystal lattice of peaks and valleys, are quantized in multiples

of as follows

2 x n =

0P (x) dx = 2

E n

0(E n V )

dxdV

dV =

2 E 1

0(E n V )

dxdV

dV + 2 E 2

E 1(E n V )

dxdV

dV +

2 E 3

E 2(E n V )

dxdV

dV + ........... + 2 E n

E n 1(E n V )

dxdV

dV =

A(n )1 + A

(n )2 + A

(n )3 + ....... + A(n )n = n ; n = 1 , 2, 3,.... (3.2)

For a given value of n = 1 , 2, 3, ..... the sum of each one of these n- aperiodic-

crystal-like bands contributes to a net value of area An = n . This is not tosay that the areas in (3.2) are equally partitioned in one unit of ! It is thewhole sum which adds up to n . For any given value of n one can take theratios of areas to obtain a sequence of fractions

A(n )1

n, A

(n )2

n, A

(n )3

n, ....... A

(n )n

n; n = 1 , 2, 3, ..... (3.3)

(one should take the magnitude of the areas in the case of negative contributionsin the integrals). It is known that the self-similarity of the Farey sequence of fractions posses remarkable fractal properties [27] that is very relevant to thevalidity of the Riemann Hypothesis (RH) based on Farey fractions and theFranel-Landau shifts [28]. Do the area-fractions (3.3) follow a Farey sequencewhen 2x(V )/ =

N (V, (V ))?

A fractal SUSY QM model to t the spectrum of the imaginary parts of thezeta zeros n was studied in [16] based on a Hamiltonian operator that admitsa factorization into two factors involving fractional derivative operators whosefractional (irrational) order is one-half of the fractal dimension ( d = 1 .5) of thefractal potential found by Wu-Sprung [13]. A model of fractional spin has been

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constructed by Wellington da Cruz [29] in connection to the Fractional QuantumHall effect based on the lling factors associated with the Farey fractions. Thisapproach based an a fractional Quantum Hall should be contrasted with theordinary Quantum Hall Effect approach to the RH [19].

The integral depicting the phase space area of a domain Dcan be writtenas a result of Stokes theorem as

DdX dP =12 C (P dX XdP ). (3.4)

where the line (contour) integral is dened along the boundary region of thedomain Dof phase space. Since the potential is symmetric V (x) = V (x)another way of obtaining the same result for the net areas is to compute theareas from the line integral (3.4) using the equality due to the symmetry of thepotential

12 C P dX = 12 X dP. (3.5)

The relation P + V = E n P = E n V yields

An = X dP = X d (E n V ) = 2 E n

0X dV =

E n

0

d N (V, (V ))dV

dV = [ N (E n , (E n )) N (V = 0 , (V = 0)) ] = n .(3.6)

Therefore, in general, one has the dual or reciprocal forms for the same phase

space area An as a direct consequence of Stokes theorem

An = 2 E n

0X (V ) dV = 2

E 1

0X (V ) dV + 2

E 2

E 1X (V ) dV +

2 E 3

E 2X (V ) dV + ........... + 2

E n

E n 1X (V ) dV =

I (n )1 + I

(n )2 + I

(n )3 + ....... + I (n )n = n ; n = 1 , 2, 3, .... (3.7)

where we have re-written x as X . Because of the relationship 2 X (V ) = d N (V, ( V ))dV derived in the previous section, and by setting N (V = 0 , (V = 0)) = 0, onehas

An = 2 E n

0X (V ) dV =

E n

0

d N (V, (V ))dV

dV = N (E n , (E n )) = n .(3.8)

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From eqs-(3.7, 3,8) one can write the values of the integrals associated to eachone of the respective intervals [0 , E 1], [E 1 , E 2], [E 2 , E 3], ......, [E n

1 , E n ] as

+ (21) + (32) + ....... + (n(n1)) = + + + ....... + = n .(3.9)which is a direct consequence of the quantization conditions (2.28) when N (V =0, (V = 0)) = 0 N (E 1 , (E 1)) = 1; N (E 2 , (E 2)) = 2; ........., N (E n , (E n )) = n. (3.10)

Therefore, from this decomposition of the areas in terms of X dV integrals,one has now an equipartition of the area An = n into n-single bits and whosequantum of area is I

(n )1 = , I

(n )2 = , I

(n )3 = , ....... I

(n )n = . (3.11)

A full cycle requires starting at , going to + and back to , thus thefull cycle will generate 2n area-bits, consistent why the n-th-winding numberof the orbit associated with the n-th zeta zero E n . This is where one canmake contact with the work by Berry and Keating [14] on the periodic orbitsassociated with the chaotic Riemann dynamics whose periods are multiplesof logarithms of prime numbers based on the classical Hamiltonian H = xp [14]and the Gutzwiller trace formula.

Now we are ready to nd the relationship between area quantization, andthe distribution of primes and zeta zeros. The following integrals Y n , for n =1, 2, 3.... also give the same values of n , after the renaming of variables y = V 2n

Y n = 12 2n0 log ( V 2n ) dV = n 2n

0log ( V

2n) d( V

2n) =

n 1

0log (y) dy = { n y ( log y 1 ) }10 = n . (3.12)

because 0 log 0 0. Upon equating the 3 integrals (3.2, 3.7, 3.12), and afterusing the results of the previous section x(V ) = 2 N (V )x (V ) = 2 N (V ),the area quantization in phase space reads

An

= E n

V =0( E n V )

d2 N dV 2

dV = E n

0

d N (V, (V ))dV

dV =

1

2 2n

0log (

V 2n

) dV = n 1

0log (y) dy = n; n = 1 , 2, 3, ...... (3.13)

The fact that the integrals are equal does not mean the integrands are equal,nevertheless one can still establish the following one-to-one correspondence of the integrands and domains of integration as follows

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12 log ( V 2n ) d N (V, (V ))dV (V ); 2 n E n . (3.14)

d N (V, (V ))dV ( E n V )

d2 N (V ; (V ))dV 2

. (3.15)

From the correspondence (3.14) one learns that the irregularly spaced zetazeroes E n has a correspondence with the evenly spaced energy levels given by2n . While the logarithmic integrand 12 log( V 2n ) = 12 log(y), which hasthe same functional form as the inverse average density of primes log(x) (up toa sign and numerical factor) has a correspondence to the density of zeta zeros(V ) in the critical line. The negative sign 12 log( V 2n ) has a connection toConnes work on the RH and Noncommutative Trace formula where the locationof the zeta zeroes were interpreted as absorption lines of the spectrum, insteadof emission lines [18] .

The prime number theorem states the the number of primes P (N ) in theinterval [0 , N ], for large N , is of the order of P (N )(N/logN ). The averagenumber density of primes is P (N )/N (1/logN ), so its inverse is log(N ). Thedensity of primes is instead dP (N )/dN = (1 /logN ) (1/logN )2 . We believethis is no coincidence for the even harmonious spacing of the energy levels 2 nis related to the imaginary parts of the zeros of :

sin (iz ) = i sinh (z) = i sinh (x+ iy) = i sinh (x) cos(y) cosh(x) sin (y) = 0 sinh (x) cos(y) = 0 , and cosh (x) sin (y) = 0 . (3.16)

The solutions to these last 2 equations is x = 0 , y = 2n . Therefore thezeroes of the function sin (iz ) = i sinh (z) = 0 are are zn = 0 i2n , whichsatisfy an analog of the Riemann Hypothesis. They all line in the vertical lineRe(z) = 0, with the main difference being that the latter zeros are all evenlyand harmoniously spaced in intervals of 2 along the imaginary axis. Thus,the E n 2n correspondence would be another reection of the irregular butharmonious distribution of the primes.

It is warranted to explore the connections to the area quantization of thequantum droplets in the Quantum Hall Effect. Studies of the Lowest LandauLevels in the quantum mechanical model for a charged particle on a plane ina constant uniform perpendicular magnetic eld by [19], have shown to yieldthe absorption level spectrum of the zeta zeros by Connes [18] and related tothe Berry-Keating [14] model of the average level density of the zeta zeroesbased on the classical Hamiltonian H = xp. It was conjectured [19] that thefluctuating part S (E ) of the counting function N (E ) (2.12) might be accountedfor by the higher Landau levels. The upshot of our results is that we have beeninvolved with the full-edged Riemann-van Mangoldt expression N (E ) in eq-(2.12) which not only has the uctuating part S (E ), but also the higher order

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O(E n ) corrections as well, the (E ) terms, in addition to the standard termsE2 (log

E2

1) + 7 / 8.

To nalize, we should add that since we are dealing with Dirac-like operatorsone must not forget the existence of anti-particles with negative energy states

E n , although we are not working in four dimensions where the CP T theoremapplies. Negative energy states is consistent with the fact that the zeta zeros inthe critical line appear in pairs of complex conjugates 12 iE n . The absence of apositive-energy electron behaves as if a positron of positive charge and negativeenergy were created. Eq-(2.1) admits the analog of negative energy in one-dimension, ( or 0+1-dim ) by simply writing the dispersion relation ( P + V )2 =

M2 (P + V ) = M= E . Therefore, the existence of E eigenvalues iscompatible with the zeros appearing in pairs of complex conjugates 12 iE nalong the critical line. In Electro-Magnetism (EM) the canonical momentum isdened by the replacement p p eA , where A is the EM potential and eis the negative charge of the electron. Thus having the generalized momentumP + V bears some relation to the canonical momentum in EM , which bringsup again the connection to the work on Landau Lowest Levels, Quantum HallEffect... by [19].

In future work we will explore the relations of our work to

Chaotic Renormalization Group Flows, Universal Mandelbrot Set, Phasetransitions, attractors, Julia sets, .... by [31]. Fractal strings, fractal membranes, noncommutative spaces, Dirac-like op-erators, spectral triples , quasi-crystals, modular ows of the moduli space of

fractal membranes, adeles, arithmetic geometries, ..... in connection to the owsof zeroes of zeta functions towards the critical line, by Lapidus et al [26]

Connes noncommutative trace formula [18]. The fermionic Trace Formula,supersymmetry, Witten index and the Mobius function [30]Black Hole entropy and area quantization in Loop Quantum Gravity; Fareysequences, fractal statistics, and the fractional Quantum Hall Effect [29]. Cyclotomy, Phase quantization, Ramanujan sums, ...... by Planat et al[25].To summarize this work, in sections 1 and 2 we have presented two plausi-

ble methods to prove the Riemann Hypothesis. One was based on the modularproperties of functions and the other on the Hilbert-Polya proposal to ndan operator whose spectrum reproduces the ordinates of the zeta zeros in thecritical line. We described in detail how the Dirac-like operator with a poten-tial V (x) in eq-(2.1) reproduces the spectrum after the boundary conditionsE (x = ) = E (x = + ) are imposed. Such potential V (x) was derivedimplicitly from the relation x = x(V ) = 2 (V ) = 2 (d N (V, (V )) /dV ). Atthe end of section 2 , we explained how to provide the implicit form of thepotential V (x) for the self-adjoint Schroedinger operator that also reproducesthe zeta zeros. Crucial in the construction, was the introduction of an energy-dependent cut-off function ( E ). In the nal section 3 the natural quantizationof the phase space areas (associated to nonperiodic crystal-like structures) ininteger multiples of follows from the Bohr-Sommerfeld quantization condi-tions of Quantum Mechanics. It allows to nd a physical reasoning as to why

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the average density of the primes distribution for very large x : O( 1logx ), hasa one-to-one correspondence with the asymptotic limit of the inverse averagedensity of zeta zeros in the critical line.

Acknowledgements

We thank Frank (Tony) Smith and Jorge Mahecha for discussions and helpwith Mathematica; to Paul Slater for bringing to my attention the work of A.Karatsuba and M. Korolev, and M. Bowers for assistance.

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riemann hypothesis from area quantization, renormalization group

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