Physical interpretation of the Riemann hypothesis

Transcription

1 Physical interpretation of the Rieann hypothesis Ditry Pozdnyaov Faculty of Radiophysics and Coputer Technologies of Belarusian State University Nezavisiosty av4 Mins Belarus E-ail: Keywords: Rieann hypothesis; Rieann zeta function; Rieann Xi function; Decaying quantu state Abstract: An equivalent forulation of the Rieann hypothesis is given The physical interpretation of the Rieann hypothesis equivalent forulation is given in the fraewor of quantu theory terinology One ore power series related to the Rieann Xi function and the Rieann hypothesis is considered Soe roots of the polynoial connected with the power series are studied It is shown that the Rieann hypothesis is true But it is undecidable and ust be considered as an axio As the Rieann hypothesis so the proble of its proof are so well-nown that writing of even a short introduction is unreasonable rewriting of copy-boo axis But it is only necessary to note that a new point of view on the hypothesis in the fraewor of physical applications of the Rieann zeta function (see for exaple [ ]) is proposed in this study Matheatics Lie Rieann let us exclude fro consideration the trivial zeroes of the Rieann zeta function () s ( it) for convenience Their existence is caused by the fact that the Euler Γ-function entering the expression for -function [4] has singular points Therefore instead of the Rieann zeta function let us consider Rieann s upper-case Xi function s s s ( t ) ( it) ( s) s ( s) x x ( x) dx ss ( ) () ( x) exp n is the theta series (theta function) The zeroes of Rieann -function evidently coincide with the nontrivial zeroes of Rieann -function [ 4] Let us introduce new variables and y ln( x) After a nuber of transforations of eq() one can obtain relation nx t i ( t ) ( ti) cos y( y) dy coszy ( y) dy ( z) () z 4 ( y) exp ( y 4) n exp( y) n ti i ti i So far as it is well nown that the zeta function has not the nontrivial zeroes outside the critical strip and on its boundaries (J Hadaard Ch J La Vallee Poussin 896) let us consider only the case when values of are in the interval fro / to / ( ) Let us apply relation cos t i exp y y dy () 4 ti i ti i for subsequent transforation of eq() It is true t and ( ) Eq() is the result of Fourier transfor of the second ter in eq() Then we get expression ( z) coszy ( y) dy (4)

2 ( y) exp y4 exp ( y 4) n exp( y) n It should be noted that identical equality g( y)exp izy dy g( y)cos zy dy (5) is true in the sense of Cauchy principal value of the integrals for every even function g Taing into account the liits of integration in eq(4) the function (y) can be naturally replaced by the function ( y ) for which the identity is true As a result we have equation ( z) expizy y dy (6) instead of eq(4) The choice of sign before the iaginary unit does not obviously influence the zeroes of - function (see eq(5)) Let us choose the lower sign for definiteness Let us change the variables t y/4 x and introduce function ( ) R( ) AR( ) R( x)exp i( i) xdx K R( x)exp( ikx) dx AR( K) R( K) (7) ( ) r ( x) x ; Rx ( ) r( x) r( x) x; (8) r ( x) x ; r ( x) exp( x) expx n exp( 4 x) n ( ) exp( ) exp exp(4 ) n r x x x n x The zeroes of R -function { Kn ( i) n n } obviously coincide with the zeroes of -function { zn ( ti ) n n } up to constant factor that is Kn zn n { } But in contrast to - function all the values of R -function are obviously the real positive nubers or zero in other words K R So basing on eqs() (8) the Rieann hypothesis is reduced to (copletely equivalent to) conjecture that and ( )\{} R > Thus the Rieann hypothesis given in such a for is a stronger assertion than the evident non-strict inequality R and ( ) If the Rieann hypothesis was false it would ean that and ( ) \ {} : R = Physics We need soe generalizations for the physical interpretation of the hypothesis Let us consider a set of wave functions {} in the coordinate representation : which are the stationary Schrödinger equation solutions describing soe bounded quantu states [5 6] Then by analogy with eq(7) we have relation ( ) A( ) ( x)exp i( i) x dx ( x)exp( ikx) dx A( K) ( K) (9) It is evident that K In context of quantu physics terinology in eq(9) K is the wave vector which is coplex in general case ( K i ; ) [5 6]; x is the coordinate (position); A is the wave

3 function in the wave vector representation [5 6]; is the function of spectral density of quantu states [5 6] The considered functions are wave functions describing quantu states of particles in case of their nonrelativistic one-diensional finite otion If = then eq(9) characterizes the expansion of a -function in the stable states of wave-vector space which is a subspace of phase space x Here x is the coordinate (position) space which is a subspace of phase space too The foralis of path integrals can be applied to describe any quantu syste with the stable (non-decaying) states [7] It is equivalent to foralis based on an evolution wave equation with a real Hailtonian [7] The quantities n (n ) satisfying the equality ( n ) = are nothing else than forbidden (non-excited) states in the wave-vector space If then eq(9) characterizes the expansion of a -function in the unstable (decaying) states of coplex wave-vector space K which is a subspace of generalized phase space x K As a rule an unstable quantu state is either the spontaneously decaying quantu state or the quantu state decaying during irreversible decoherence process The foralis of restricted path integrals can be applied to describe any quantu syste with the unstable (decaying) states [7] Under soe conditions it is equivalent to foralis based on an evolution wave equation with a coplex Hailtonian [7] Thus during passage fro ( = ) to K \ ( ) we are passing on fro consideration of physical systes with the stable quantu states to consideration of physical systes with the unstable ones In particular the passage is equivalent to passage fro consideration of isolated quantu systes to consideration of open ones [7] It is easy to show (see Appendix A) that the function R(x) can be considered as the wave function Let us call it as the Rieann wave function (the Rieann wave function in the coordinate representation) Let us also call the function A R (K) as the Rieann wave pacet (the Rieann wave function in the wave vector representation) and the function R( K) AR( K) as the Rieann spectral function (function of spectral den- sity of quantu states) Let us now appeal to a siple but significant exaple of finite otion of a particle that will sufficiently illustrate the disappearance of forbidden states in the spectru at passage fro the expansion of - functions in the real wave vectors to the expansion of -functions in the coplex wave vectors K = i In particular let us consider a particle in an infinitely deep potential well of width a for which the states are defined by the wave functions [5 6] acos n x a x [ a a ] n ; n( x) asin n x a x[ a a ] n 4 ; x [ a a ] n 4 In that case the spectral density calculated by eans of eq(9) is given by expression a n a 4 n ; 4n cos ( a )cosh ( a ) sin ( a )sinh ( a ) 4n sin ( a )cosh ( a ) cos ( a )sinh ( a ) a n a 4 n( ) n 4 ; 4 n cos ( a ) na n ; a n a 4 n sin ( a ) na n 4 ; a n a a na n 4 4

4 It is evident that \{} n n > and n n( a ) at n The function n ( ) vanishes in an infinite nuber of points lie the function R ( ) So taing into account everything entioned above an equivalent forulation of the Rieann hypothesis in the fraewor of quantu theory terinology can be laid down for exaple lie this: the Rieann spectral function vanishes only for the real values of wave vector Soe ore atheatics Expanding the cosine in eq() in the Taylor series the first ter in the equality can be represented by a power series That is eq() has turned into the following equality ( z) c z () z 4 ( ) c y exp ( y 4) n exp( y) dy 4 ( )! () n According to ref [8] the iproper integral in eq() is reduced to the corresponding order derivative of the incoplete Gaa-function Naely we have the expression 4 exp ( 4) exp( ) ln ( )exp y y n y dy t t n t dt d x n x n dx x4 And as a result we coe to the forula ( ) d c exp ln( ) x n x n 4 ( )! x 4 n dx () According to Cauchy Hadaard theore the series in eq() converges z In particular anyone can be convinced that ln( ) (4 ) ( ) c e c at It is evident (see eq()) that zeros of (z) copletely coincide with zeros of the power series ( ) a z P ( z) z 4 () 4 ( )! d a exp ( y 4) ln(y) n exp( y) dy exp xln( n ) x n x 4 n n dx (4) Taing into account the explicit for of eqs() () and () the Rieann hypothesis is also equivalent to one ore assuption: all the roots of equation P ( z) (5) are the real nubers Let us further consider the following polynoial M ( ) a z PM ( z) z 4 (6) 4 ( )! M which relates to the power series P (z) For large values of M as is nown [9] zeroes of P M (z) can be found only by eans of nuerical ethods In figs and the roots z n of the polynoial equation P ( ) M z (7) are represented for soe values of M 4

5 Fig All roots of eq(7) in the coplex half-plane Re (z) > nuerical exact The nuerical error in finding the roots is less than ie z z z n n n Fig All real positive roots of eq(7) The nuerical error in finding the roots is less than 4 nuerical exact 4 ie z z z n n n 5

6 It is evident fro the figures that in contrast to the real roots of eq(7) there is no disorder in distribution of the coplex roots They are strictly ordered Iportant rears In accordance with ref [4] the Rieann Xi function (s) and therefore the function (z) is not represented by an analytical expression in closed for ie it is no way to pass fro the integral in eq() to an expression with a finite nuber of operations over a finite nuber of eleentary functions (the integral is not represented by quadratures) As a result there is no analytical way to find and verify the roots of eq() Also there is no analytical way to find and verify the roots of eq(5) because of both the ipossibility to represent coefficients a by quadratures and the infinite nuber of ters in eq(5) [9] In that case it is ipossible to chec analytically the belonging of every root z n to or \ ( z n ) Finally the only way of such a search and exaination is a nuerical one But because of the infinite nuber of roots of eqs() and (5) [4] any nuerical search and exaination of the all is ipossible fro the practical point of view since it taes infinitely long coputational tie Consequently we have two situations The first one corresponds to the case when the Rieann hypothesis is true Then obviously every found root is a real nuber At that the rest part of the roots which is always infinite has never been checed It follows fro this that if the hypothesis is true then it is ipossible to prove it Naturally the hypothesis negation cannot be proved for such a case in principle The second situation corresponds to the case when the Rieann hypothesis is false and consequently it is unprovable in principle but its negation is provable since there is an algorith which will find at least one root z \ sooner or later On this basis we can conclude that the Rieann hypothesis can only be either consistent and undecidable or decidable and inconsistent According to ref [] the Rieann hypothesis is consistent and therefore it is undecidable As a result it is true according to Gödel s incopleteness theores For such a case we could draw a parallel with the Euclidean geoetry (see Euclid s fifth postulate: the parallel postulate) when considering the Rieann hypothesis as the fifth postulate of atheatics sui generis In particular Gödel s incopleteness of a priori incoplete axios syste in which the Rieann hypothesis is forulated could appear itself just through the hypothesis At that the syste could be copleted if to consider the hypothesis as the stateent ie issing postulate or axio and not the assuption Conclusions Thus the equivalent forulation of the Rieann hypothesis is given The physical interpretation of the Rieann hypothesis equivalent forulation is given in the fraewor of quantu theory terinology The power series P (z) related to the Rieann Xi function and the Rieann hypothesis is considered Soe roots of the polynoial P M (z) connected with the power series P (z) are studied It is shown that the Rieann hypothesis is undecidable and true according to Gödel s incopleteness theores Acnowledgeent The author is very grateful to Dr Serguei K Seatsi (Laboratory of Physics of Living Matter Ecole Polytechnique Fédérale de Lausanne Lausanne Switzerland) for helpful discussions and a critical review of the paper Appendix A R(x) (see eq(8)) is the wave function as it satisfies the basic requireents of quantu echanics (the regularity conditions) [5 6] and it is the eigenfunction of the Schrödinger Hailtonian Let us show this below R(x) is evidently single-valued R(x) is evidently finite R(x) is noralizable since Rx ( ) dx exp x dx 4 The function R(x) and its first derivative R'(x) are obviously continuous x \{} (see eq(8)) R(x) and R'(x) are also continuous in the point x = since li R( x) exp( n ) R() x n x li R( x) R() 6

7 Anyone can be convinced of this fact applying the well-nown relation for the theta series (x) [4] 4 () () 5 It is easy to be convinced that the function R(x) satisfies the Schrödinger equation R( x) ur( x) R( x) RR( x) u ( x) 6 R R exp4 exp6 exp4 n n x x n x n n exp x n exp 4 x Thus R(x) is the wave function describing a bound state Naely the ground quantu state in the potential well characterized by the potential function u R Appendix B It is extreely iportant and interesting to search and investigate stochastic processes of a certain ind which apparently have not been studied yet The processes are non-marovian stochastic processes described by the following autocorrelation function t exp tt n exp 4 tt n () t exp exp( n ) n References [] Sierra G A physics pathway to the Rieann hypothesis 8 () [] Schuayer D Hutchinson DAW Colloquiu: Physics of the Rieann hypothesis Rev Mod Phys 8 7 () [] Dettann CP New horizons in ultidiensional diffusion: The Lorentz gas and the Rieann hypothesis J Stat Phys () [4] Titcharsh EC The Theory of the Rieann Zeta-Function (Oxford University Press New Yor 986) [5] Davydov AS Quantu Mechanics (Pergaon Press Oxford 99) [6] Landau LD Lifshitz EM Quantu Mechanics: Non-Relativistic Theory (Butterworth-Heineann Asterda- Boston ) [7] Mensy MB Quantu Measureents and Decoherence: Models and Phenoenology (Kluwer Acadeic Publishers Dordrecht ) [8] Gradshteyn IS Ryzhi IM Table of Integrals Series and Products (Acadeic Press 7) [9] King RB Beyond the Quartic Equation (Birhäuser Boston 996) [] Perna T On an essential connection of the Rieann hypothesis and differential equations Asian J Math Appl 4 9 (4) t t 7