Zeta Regularization Applied To The Problem Of Renormalization And Riemann Hypothesis

ZETA REGULARIZATION APPLIED TO THE PROBLEM OF RIEMANN HYPOTHESIS AND THE CALCULATION OF DIVERGENT INTEGRALS Jose Javier Garcia Moreta Graduate student of Physics at the UPV/EHU (University of Basque country) In Solid State Physics Address: Address: Practicantes Adan y Grijalba 2 5 G P.O 644 48920 Portugalete Vizcaya (Spain) Phone: (00) 34 685 77 16 53 E-mail: [email protected]

MSC: 45G05, 47H30 , 03.65.-w , 11.10.Gh

ABSTRACT: In this paper we review some results of our previous papers involving Riemann Hypothesis in the sense of Operator theory (Hilbert-Polya approach) and the application of the negative values of the Zeta function

 (1  s )



to the divergent integrals

x

s 1

dx and to the

0

problem of defining a consistent product of distributions of the form

D m ( x) D n ( x) , in this

paper we present new results of how the sums over the non-trivial zeros of the zeta function

 h(  ) can be related to the Mangoldt function  

0

( x) assuming Riemann Hypothesis.

Keywords:Zeta regularization, Urysohn equation , exponential nonlinearity , Riemann Hypothesis Hilbert-Polya operator, divergent integral

1.Spectral Zeta function  H ( s) and Riemann Hypothesis : In case Riemann Hypothesis (RH) is true, in a previous paper [6] we give the physical equivalence between the explicit formula for the Chebyshev function  0 ( x) and the ˆ formula for the trace of the Unitary operator Uˆ  eiuH , where H is the Hamiltonian 1  operator    iHˆ  n  0 , that is H is precisely the Hilbert-Polya operator solution to 2  Riemann Hypothesis , let be the integral representation

 x   (0) 1  log(1  x 2 ) x  1 1 ci  ( s ) x s  x    (0) 2    0 ( x)    ds   2 i ci  ( s ) s  x 1  0

1

(1.1)

Letting x  eu , and differentitating with respect to ‘u’ we find the (trace) identity

eu / 2  e  u / 2



 d  0 (eu ) eu / 2 ˆ  3u u   eiuEn  Tr Uˆ  eiuH du e e n 



u >0

(1.2)



Using the semiclassical representation for the trace

e

iuEn

in terms of an integral over

n 

Phase Space , we have that the potential V(x) inside Hamiltonian H can not be arbitrary but must satisfy a kind of nonlinear Urysohn integral equation ( r > 1) 

r

iV ( x )

dx 



log(r )  d  0 (r ) 1   i / 4 1  3 e 1  dr r r  r 

r  eu

(1.3)

d  0 ( x) 1    ( x  p ) dx log( x) p , p (sum taken over prime and prime powers ) .However (1.3) is too complex to have a known analytic solution, a good method to solve would be to suppose that the Operator proposed by Berry and Keating [2] plus an interaction is the correct Hilbert-Polya operator, in that case H b  xp  W ( x) and we can linearize (1.3) at first order in the coupling constant ‘  ’ as The derivative of the Chebyshev function is defined as

 

Tr eiuH  ˆ

 2  iu  dpFˆ W ( x), u  |u |

 Fˆ W ( x), u   dxeiuxpW ( x) 

Also, if we introduce the function Z u   



 dxe

iu (V ( x )  x )

(1.4)

, with continuos partial



derivatives  k Z u ( ) , then solving (1.3) is equivalent to finding a solution to the initialvalue problem

Z u ( )  

  Z u ( ) k k   iu   d k Z ( )  0 k k  u   k 0 (iu )   (1.5)

 u / 2  u / 2 d  0 (e )  u e  3u u  e e e du e e    u

u/2

 i 4

 Z u (0)

Expression (1.8) tells us that proving RH is equivalent to show that the ODE given in  1 d kV ( x) (1.5) with d k   R and d k  , V ( x )  d k x k using (1.5) together with  x 0 k k ! dx k 0 a finite power expansion for V(x) , using (1.5) we could obtain the constants d k   R to get an approximate solution for the potential V(x).

2

1  If RH is true and    iEn   0 , with En   E n being the eigenvalues of a certain 2  2 operator H  p  V ( x) , using expression (1.2) and the functional equation s   (1  s )  2(2 ) s Cos   ( s ) ( s) ,then for n  0 we can define an spectral Zeta  2  function , involving the nontrivial zeros of Zeta and primes and prime powers

s  Sec    t 1 1   s  (s) 2   itHˆ t/2   t d  0 (e )     2   3t t  t s 1 dtTr e dte 1  e  s   0  (1  s) 0 2 ( s ) dx e e  n 0 En  (1.6)

 





The value

E n 0

n

e



d  (0) ds

would be the regularized product of all the positive

‘Eigenvalues’  En  this expression can also be used to obtain a Dirac measure for the En , let us introduce    1  1   s1 s  E dt     t    t  n  n 0 0  n0 En  En  

  (s)   dtK 0 (t )t s1  (1  s ) 0



(1.7)

Using the properties of the Mellin transform applied to solve linear integral operators 

I [ f ]   dtR ( xt ) f (t ) , if we combine (1.6) and (1.7) we get the result 0

  1  1  dt e 1/ 2t d  0 (e1/ t ) e1/ 2t  ( x)     x     K 0 (2 xt )  e1/ 2t  2  3/ t 1/ t  En  0 t t dt e e  n 0 En   

(1.8)



If we took the Mellin transform  dxx s 1 inside (1.8) together with the change of 0

variable xt=z we would recover equation (1.6) , note that the Mellin transform of the 1 Kernel K 0 (2 xt ) does not depend on the nontrivial zeros    it . 2 Using test functions

1 1 h  x 2

i  inside (1.8) obtained from our Trace formula for x

  we can relate the convergent sum  h(  ) to a sum over primes and prime ˆ

Tr eiuH



powers

3

 K 0  2 xt   1/ 2 t e1/ 2 t d  0 (e1/ t ) e1/ 2 t  1 i   3/ t 1/ t   c.c   h(  ) 0 dxh  2  x  0 dt xt  e  t 2 dt e e   1

(1.9) Formula (1.9) and its result can be compared with sums Chebyshev function) and Z (n)   

  ˆ

a   (explicit formula for

1 n  N , that can be calculated exactly. n

o The Trace Tr eiuH and the sum

 h( )

Even though we can not solve equation (1.3) we can use the Trace expression (1.2) to find stimates for sums  h( ) . First we define a couple of function g(x) and h(x) with 

the following properties   



Both g(x)=g(-x) and h(x)=h(-x) are even functions g ( x) lim exists and it is finite x 0 x The functions h(x) and g(x) are related by a Fourier Cosine transorm  1 dxh( x)Cos ( x)  g ( )  0 The function h(x) can be defined by analytic continuation to the region of complex plane defined by Re (s) =0 , in particular h(i / 2) 1  i

If RH (Riemann Hypothesis) is true, then the Trace (1.3) is just a sum of cosines  2cos( u ) , then if we take g(u) as a test function  0



 

iuH  dug (u )Tr e  ˆ

0



h(i / 2)  h(i / 2)   (n) g (u )   2 h( ) g (log n)   due  u / 2 2u  2 e 1 n n 1  0 0 (1.10)

In order to obtain (1.10) we have used the representation in terms of Dirac deltas of the  d  0 ( eu )   ( n )  derivative of Chebyshev function to get  dug (u ) g (log n) , and the du n n 1 0 

Euler formula for cosine to represent the integral  dug (u )eu / 2 as the sum 0

1  h(i / 2)  h(i / 2)  . An special case is whenever we choose 2  h( x)   ( s  x)   ( s  x)  and g (u )  cos(u ) 2

4

(1.11)

Then we can use the functions in (1.11) and the formula (1.10) to get 1   '(1/ 2  is )  '(1/ 2  is ) 



1

  (s   )  2   (1/ 2  is)   (1/ 2  is)   1  4s   

2



n 0

(2n  1/ 2)

 2n  1/ 2  s  2

2

(1.12) (1.12) is the ‘density of states’ in QM , and can be used to know how many zeros of the T   form ½+is are with imaginary part less that a given ‘T’ since N (T )   ds    ( s   )    0  0 A formal derivation of (1.12) can be obtained considering the following indentities for divergent series involving zeta regularization or analytic continuation 

 ean  n 0



1 (linearity) 1  ea

 ( n)

n n 0

1/ 2is



 '(1/ 2  is )  (1/ 2  is )

 ( n) 1   (n) is   (n)  is  cos( s log n )  n  n    2  n0 n n n n 0 n 0 

(1.13)



(1.14)



And the Laplace transform of Cosine  dte st cos(at )  s ( s 2  a 2 )1 .The poles inside 0

(1.14) are of three kinds s   n from the Non-trivial zeros of Zeta function ,

s  i / 2 due to the divergent value  (1) and s  i 1/ 2  2n  n  N form the trivial zeros of zeta function -2,-4,-6,..........

o Riemann-Weyl formula and a solution for the inverse of potential V(x): A similar formula to (1.10) had been previously introduced by Weyl in 1972 [9]  ( n) 1 g (log n)  2 n n 1 

 h( )  h(i / 2)  h(i / 2)  g (0) log   2 



' 1

ir 

 h(r )   4  2  dr



(1.15) Weyl summation formula can be used to solve equation (1.10) if we make inside this integral equation the change of variable x  V 1 ( ) , then (1.10) is simply proportional to the inverse Fourier transform of proportional to another sum



e iu i / 4iu on the interval  0,  which is just  u

1

involving the imaginary parts of the Riemann   zeta zeros , to get rid off the sum we ca use (1.15) to express the inverse of potential

5

V 1 ( )  

E

A 2  i

 BCos (c   / 4)  D 

 (n)cos( log n   / 4) n log n

n 1

1/ 2

    1 ir    1 1  Fp.v   dr           4 2     r 1/ 2   r 1/ 2      (1.16)

Where A, B, c, D, E , F  R are real parameters that define the potential , from formula (1.16) the Hamiltonian would be self adjoint and its energies (imaginary part of zeros) would be real numbers. The sum involving  (n) can be treated using fractional calculus and zeta regularization 

2

 (n)cos( log n   / 4) n log n

n 1

1/ 2

 i

d 1/ 2   '(1/ 2  i )  d 1/ 2   '(1/ 2  i )  i       (1. d 1/ 2   (1/ 2  i )  d 1/ 2   (1/ 2  i ) 

17) At the points    the inverse of potential becomes  , as we can expect from

   

1/ 2

     0

1/ 2

   

1/ 2

 0

2. Zeta regularization for divergent integrals: Given the function f ( x)  x m , we can use the Euler-Maclaurin summation formula to 

obtain a recurrence relation between an integral of the form I (m, )   p m dp m Z  0



with m  x m1dx   m and the series 0

1

1

i

m

, ref [7]

i 0



B2 r amr (m  2r  1) I (m  2r ,  ) r 1 (2 r )!

I (m, )  (m / 2) I (m  1, )   i m   i 0

(2.1) 

m  x dx  0





 m m1 B2 r m!(m  2r  1)! m2 r  (  )  x dx  m    x dx 20 r 1 (2 r )!(m  2 r  1)! 0

The coefficients amr 



(m  1) vanish when m  2  2r , hence the sum inside (m  2r  2)

(2.1) is finite if m is an integer , in the physical limit the cutoff    , this makes the

6

1

series

i

to be divergent for m  1 , in this case we should use the Functional

m

i 0

equation for the Zeta function to obtain the (Regularized) value 1

lim  n m 1  2m  3m  ...   m   R (m)   (m)



(2.2)

n 1

(2.2) is the Zeta-regularized value for the divergent sum envolved in (2.1) , using this method we can compute the divergent integrals I (m,  )    , for m=1,2,3 I (0, )   (0)  1/ 2 I (1,  ) 

I (0,  )   (1) 2 (2.3)

1   B I (2,  )   I (0,  )   (1)   2 a21I (0, ) 2   2 B 3 1  I (3,  )    I (0, )   (1)   2 a21I (0,  )    (3)  B2 a31I (0, ) 2 2 2  The case m=0 is just equal to the divergent series 1+1+1+1+1+1+1+1+1+... taking the regularized value -1/2 evaluated from  (0) For an arbitrary function f(x) so its integral would diverge as a power of the cutoff  N 1 we could expand f(x) into a Laurent series convergent for |x| <1 and |x| >1 so we find 

N

N a



a

r 0

i 0 0

j 2

1 i  j 1  dxf ( x)   cr I (r , )  c1I (a, 1, )  O( )    dx  ci x    c j a

ci  


(2.4)

, taking    , and using (2.1) (2.2) (2.3) to regularize the divergent 

integrals I (m, ) we could obtain a regularized (finite) value for the integral  dxf ( x) , 0



dx can not regularized by our x a formulae, the solution would be to use the Euler-Maclaurin summation to approximate the divergent integral by a divergent Harmonic sum that can be attached a ‘Ramanujan a 1 sum’    (  =Euler-Mascheroni constant) n 1 n

however the logarithmic divergent integral I (a, 1, )  

7

o Zeta regularized product of distributions: 

Formulae (2.1-2.3) can be used to compute divergent integrals of the form

x

s 1

dx , but

0

also could give an answer to the problem of multiplication of two distributions involving Dirac delta and its derivatives D m ( x) , if we tried to define the product of distributions involving delta functions we could use the ‘convolution theorem’ applied to the Fourier transform ( A=normalization constant) : (2 ) 2 i mn D m ( ) D n ( )  F  x m  x n   AF







dtt m ( x  t )n



(2.5)

Unfortunately (2.5) makes no sense , the integral is divergent for every real or complex value of ‘x’ , if m and n are positive integers using the Binomial expansion n n i mn D m ( ) D n ( ) =    i m k AD nk ( )(1)k i nk D m k (0)[ R ] k 0  k 

i mn D m ( ) D n ( ) =

(2.6)

 n nk k n k m k AD  (  )( 1) i ( 1) 1       x mk dx    k k 0   0 n

(2.7)

‘R’ stands for regularization (regularized value) , the divergent integrals come now from the dirac delta and its derivatives evaluated at x=0 , which are proportional to 

 x dx for k=2r+1 (Odd) the integral considered in Principal Value k

is 0 , for k=2r



(even integer) the integral can be written as i 2 r D 2 r (0)  2 I (2r , )





, I (2, r )   x 2 r dx (r=integer) and can be evaluated using (2.1) and (2.2) . 0

The expression (2.7) is real ,this is what one would expect since the product of two distributions taking only real values must be real , however (2.6 ) is not still invariant under the change m  n and n  m (this is a mistake we made in paper [7] ) so we should take a more symmetrical product of distributions defined by

D   D   m

n

R

( ) 



1 m D  ( ) D n ( )  D n ( ) D m ( ) 2



(2.8)

The simplest case is m=n=0 so     R ( )   A ( ) For the case of ‘m’ and ‘n’ not being an integer or we have a shifted dirac delta D k ( x  a ) , we could use the identies for the k-th power of ‘x’ or the traslation d operator e D and D  in the form dx

8



e aD D r ( x)   (1) j j 0

In case of integrals on

 r k D r      D  1 k 0  k 

a j r j D  ( x)   ( x  a) j!

Rd

(2.9)

 dkF ( k ) , if the function F , is invariant under Lorentz 

Rd

transformations, then making a Wick rotation to imaginary time t  it ,the metric becomes ds 2  dx 2  dy 2  dz 2  dt 2 which is invariant under rotations, taking 4-

 d /2  drf (r )r d 1 , if (d / 2) 0 not we could replace the integral over the cross section (angles) d by a discrete sum dimensional polar coordinates our integral can be evaluated as 

  drf (r ,  )r

d 1

i

i

, with ‘d’ equal to the dimension of space-time

0





Example:

 dx a

x2 with    in this case the integral has a power-law 1 x

(quadratic) divergence  2 , a >1 and integer (this is not relevant since the integral diverges only for big ‘x’ ) , the Laurent series for |x| bigger than 1 is 

x  1  x 1   (1) j x1 j , if we approximate the logarithmic divergent integral j 3



of 1/x by the divergent series

1

na

(after a change of variable x=t+a) then,

n 0

the approximate ‘Zeta regularized’ value of the integral would be   (0)  '(a ) x2  a 2  (1) j 2 j        dx  ( 1) a a      1 x 2 ( a ) 2 j 2 3 j  a R

(2.10) 

Another example without a logarithmic divergence , would be  dx a

x4 in this case (1  x 2 )

n(1) 32 n , the logarithmic derivative a n2 2n  3 of Gamma function inside (2.10) is just the Ramanujan resummation of the Hurwitz series  H (1, a) avoiding the pole at s=1 

the regularized finite value is just  (0)  a  

n

Appendix A: an integral Trace for the Green function 

A formula for the sum

  ( E  E ) in terms of the Trace of the ‘Resolvent’ (green n 0

n

function ) of a Quantum Hamiltonian Hˆ n  Enn can be defined as:

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

Tr G ( x, x ', E )   4 d 4 xG ( x, x, E )     ( E  En ) R

G ( x, x ', E ) 

n 0

1 ¨ E  i  Hˆ

(A.1)

One of the easiest method to prove this , is to consider that given a convergent series    a  with sum S and its Borel transform B(s) defined by B( S , an )   dt   n x n e t then  n 0 n !  0 

S=B(S) , S   an in this case if we take the series n 0

  (1) n (i  H ) n x n    t (1i  Hˆ ) 1 E 1   E 1      dte E  i  H 1  (i  H ) E n n!  n 0  0 E

(A.2)

Where  is an small number so   0 , then using the formula for the Principal value 1 1 , in this case taking the trace of the operator inside (A.2) we P.V     ( x)  x  i x can give a proof to (A.1) using the technique of Borel resummation. 

Another example of the method of Borel resummation , let be P( x)   (1) n  (n) x n n 0

the generating function of the coefficients  (n) , let be the function f(t) defined by 

 ( s  1)   dtf (t )t s 1 then using again the Borel-generalized resummation 0





 f (t )    P( x)   dt   (1) n ( xt ) n  f (t )   (1) n  (n) x n or P( x)   dt 1  xt  n 0  n 0 0 0

(A.3)



If we took the Mellin transform on both sides  dxx s 1 we Would find 0

Pˆ ( s )  Kˆ ( s ) Fˆ (1  s ) , or in terms of improper integrals 

    ( s )  (n)( x) n   dt 0    sin( s ) n 0



since

 dtf (t )t

s

  ( s)

(A.5)

0

This last formula is known as ‘Ramanujan Master theorem’ , note that we have proved this only using the fact that for a convergent series its sums and Borel transform must be equal S=B(S).

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References: [1] Apostol, T. M. “Introduction to Analytic Number Theory”. New York: SpringerVerlag, 1976. [2] Berry, M. V. and Keating, J. P. " and the Riemann Zeros. “ In Supersymmetry and Trace Formulae: Chaos and Disorder” (Ed. I. V. Lerner, J. P Keating, and D. E. Khmelnitskii). New York: Kluwer, pp. 355-367, 1999. [3] Conrey, J. B. "The Riemann Hypothesis." Not. Amer. Math. Soc. 50, 341-353, 2003. available at http://www.ams.org/notices/200303/fea-conrey-web.pdf. [4] Delabaere E., “Ramanujan's Summation, Algorithms Seminar” 2001–2002, F. Chyzak (ed.), INRIA, (2003), pp. 83–88. [5] Elizalde E. ; “Zeta-function regularization is well-defined”, Journal of Physics A 27 (1994), L299-304. [6] Garcia J.J “Chebyshev Statistical Partition function : A connection between Statistical Mechanics and Riemann Hypothesis “ Ed. General Science Journal (GSJ) 2007 (ISSN 1916-5382) [7] Garcia J.J “ A new approach to renormalization of UV divergences using Zeta regularization techniques “ Ed. General Science Journal (GSJ) 2008 (ISSN 1916-5382)

[8] Polyanin, A. D. and Manzhirov, A. V., “Handbook of Integral Equations”, CRC Press, Boca Raton, 1998. [9]

Weyl, A. "Sur les formules explicites de la théorie des nombres", Izv. Mat. Nauk (ser. Mat.) 36 (1972)

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