A FUNCTIONAL DETERMINANT EXPRESSION FOR THE RIEMANN XI-FUNCTION

Transcription

1 To appear i Advaces i Mathematical physics A FUNCTIONAL DETERMINANT EXPRESSION FOR THE RIEMANN XI-FUNCTION Jose Javier Garcia Moreta Graduate studet of Physics at the UPV/EHU (Uiversity of Basque coutry) I Solid State Physics Addres: Practicates Ada y Grijalba 5 G P.O Portugalete Vizcaya (Spai) Phoe: () josegarc@yahoo.es MSC: 34L5, 34L5, 65F4, 35Q4, 8Q5, 8Q5 ABSTRACT: We give ad iterpretatio of the Riema Xi-fuctio ξ ( s) as the quotiet of two fuctioal determiats of a Hermitia Hamiltoia H H. To get the potetial of this Hamiltoia we use the WKB method to approximate ad evaluate the spectral Theta fuctio Θ ( t) exp( tγ ) over the Riema zeros o the critical strip < Re( s) <. Usig the WKB method we maage to get the potetial iside the Hamiltoia H, also we evaluate the fuctioal determiat det( H + z ) by meas of Zeta regularizatio, we discuss the similarity of our method to the method applied to get the Zeros of the Selberg Zeta fuctio Keywords: Riema Hypothesis, Fuctioal determiat, WKB semiclassical Approximatio, Trace formula,bolte s law, Quatum chaos.. Riema Zeta fuctio ad Selberg Zeta fuctio Let be a Riema Surface with costat egative curvature ad the modular group PSL(, R ), Selberg [4] studied the problem of the Laplacia y + Ψ x y E Ψ x y E + k x y 4 (, ) (, ) () These mometa k are the o-trivial zeros of the Selberg Zeta fuctio, which ca be defied by a Euler product over the Geodesic of the surface i a aalogy with the Riema Zeta fuctio ζ ( s) Z( s) N( P) ( p ) ( s+ k ) s () P k

2 Selberg also studied a Trace formula which relates the Zeros (mometa of the Laplacia ) o the critical lie Z + ik ad the legth of the Geodesic of the Surface i the form µ ( D) l N( P) h( k ) tah (l / / 4 dkkh k k + g N P (3) N( P) N( P) P p. p. o Here, p.p.o meas that we are takig the sum over the legth of the Geodesic, h( k ) is a test fuctio ad g( k ) is the Fourier cosie trasform of h( k ) g( k) dxh( x)cos( kx) µ ( D) is the area of the fudametal domai describig the Riema surface. I case we had a surface with the legth of the Geodesic l N( P) l p for p o the secod side of the equatio a prime umber,the the Selberg Trace is very similar to the Riema-weil sum formula [] γ i Λ Γ ' is h( γ ) h g()l g(l ) + dsh( s) + Γ 4 (4) This formula (4) related a sum over the imagiary part of the Riema zeros to k l p p aother sum over the primes, here Λ with k a positive otherwise iteger is the Magoldt fuctio, i case l N( P) l p both zeta fuctio of Selberg ad Riema are related by ζ ( + s ) ad their logarithmic Z( s) l N( P) derivative is quite similar if we set the fuctio Λ geodesic ( P) N ( P ) ζ ' s Z ' ( s) Λ ( s) Λ geodesic ( P) N( P) ζ Z s (5) P p. p. o I both cases the Riema ad Selberg zeta fuctios obey a similar fuctioal equatio which relates the value at s ad -s s / µ ( D) Z( s) exp vta( v) dv + c Z( s) 4 ζ ( s) Χ ( s) ζ ( s) (6) The costat of itegratio c is determied by settig s /, ad s s Χ ( s) ( ) Γ( s)cos for the case of the Riema zeta fuctio. With the aid of the Selberg Trace formula (3), we ca evaluate the Eigevalue y + staircase for the Laplacia ( x y )

3 p µ ( D) N E + p dkkh k k + Z + ip 4 E E 4 tah ( ) arg (7) Here p E, we ca imediatly see that the smooth part of (7) satisfy 4 µ ( D) Weyl s law i dimesio Nsmooth ( E) E, the oscillatory part of (7) satisfy 4 Bolte s semiclassical law [4] (page 34, theorem. ) λ arg Z + i E with λ, the brach of the logarithm iside (7) is chose, so arg Z i this case the Selberg Zeta fuctio is the dyamical zeta fuctio of a Quatum system ad the Eergies are related to the zeros of Z( s ).. A fuctioal determiat for the Riema Xi fuctio ξ ( s) From the aalogies betwee the Riema Zeta fuctio ad the Selberg Zeta fuctio, we could ask ourselves if there is a Hamiltoia operator (the simplest secod order differetial operator which has a classical ad quatum meaig ad it is well studied )i the form d Ψ ( x) H Ψ ( x) + V ( x) Ψ () x E Ψ x Ψ Ψ E γ (8) dx s / s So for the Riema Xi-fuctio ξ ( s) s( s ) Γ ζ ( s) ξ ( s) we have that ξ i E f ( x) x > + N, the potetial is give by V ( x), at x x there is a ifiite wall so the particle iside the well ca ot peetrate the regio x <. For the case of the Hamiltoia (8) the exact Eigevalue staircase is [9] ϑ( E ) N( E) H E E arg + i E + arg + i E + ( ) ξ ζ (9) With x> H ( x), x< T T T T ϑ( T ) arg Γ + i l l e 8 48T Also we will prove how the Riema Xi fuctio ξ ( s) is proportioal to the det H s( s), ad how the desity of states ca be fuctioal determiat 3

4 evaluated from the argumet of the Xi-fuctio E d I mlog det ( H + iε p ) ρ( E) δ ( p γ ) p dp γ p As a simple example of how Quatum Mechaics ca help to solve problems of fidig the roots of fuctios, let be a particle movig iside a ifiite potetial well, the eergy is give by E p ad the oe dimesioal Schröediger equatio [7] i uits h m ( h is the reduced Plack s costat with value 34 h.5. J. T ) d u( x) + u () () H u x V x u x E u x u E dx u x A x, i this case the Euler s product formula for the sie fuctio si is the quotiet betwee fuctioal determiats si ( x ) x det ( H x) x E det ( H ) H H () We ca also compute the desity of states to get the Poisso sum formula ρ( E) δ E E δ p + δ p e p p i p ( ) () o Zeta regularized determiat for ξ ( s) : Give a Operator P with real Eigevalues { E }, we ca defie its Zeta regularized determiat [6] i the form d ds ζ det P + k exp P ( s, k ) s () Here (, ) { s ζ P s k Tr P + k } ( E + k ) s is the Spectral Zeta fuctio of the operator take over all the Eigevalues, the relatioship betwee this spectral zeta fuctio ad the Theta fuctio Θ ( t) exp( te),t> always, is give by the Melli trasform dt e tk s s t t Θ ( E ) ( s) t + k Γ. If P is a Hamiltoia Θ t te we ca obtai the Theta fuctio exp (approximately) by a itegral over the Phase space [7] 4

5 Θ ( t) exp te dp dxe dxe Θ ( t) t tp tf ( x) tf ( x) ( ) WKB (3) The expressio (3) depeds oly o the mometum ad the fuctio f ( x ) defied i (8) to evaluate the Theta fuctio, if we combie (3) ad the defiitio of the Theta fuctio for the Eigevalues Θ ( t) exp te s dtn( t) e dx dp exp( tp tf ( x)) st ( ) (4) tf ( x) tr dv ( r) (5) dq dp exp( tp tf ( x)) dxe dre t t dr From expressios (4) ad (5) ad settig N () (after chages of variable) / sx sx d / dx (6) s dxn( x) e dxf ( x) e f ( x) N( x) To prove (6) we have used the properties of the itegral represetatio for the iverse Laplace trasform c+ i α st α D f ( t) dsf( s) e s α kt α kt i D e k e α R (7) c i Ad the fact that if two Laplace trasforms are equal the L{ f ( t) } L{ g( t) } implies that f ( t) g( t), for the case of the Riema Zeros N ( E ) argξ i E + (Bolte s semiclassical law i oe dimesio) so / d f ( x) argξ i x / +, sice we wat our potetial iside (8) to be dx positive wheever we take the iverse we must choose the POSITIVE brach of the iverse i order to get f ( x) o the iterval [, ), the half derivative ad the half itegral for ay well behaved fuctio are give i [3] / x d f ( x) d dtf ( t) / dx Γ (/ ) dx x t / x d f ( x) f ( t) dt / dx Γ (/ ) x t (8) We have writte implicitly the potetial iside (8), if the fuctio f ( x ) is defied by the fuctioal equatio / d H ( x γ ) f ( x) argξ i x / +, the we may evaluate the dx x γ Spectral Zeta fuctio of the Quatum system give i (8), the 5

6 det H + z d d ξ ( z / ) exp ζ P ( s, z ) s + ζ P ( s,) + s det H ds ds ξ (/ ) (9) / d f ( x) argξ + i x / For the potetial defied by, we ca evaluate dx df ( x) the Theta kerel uisg (5) ad (6) e te tx Θ t dx e t, for dx this potetial the spectral theta fuctio ad its derivative are ζ d ζ γ ζ γ s γ ds + z () H ( s, z ) - H (, z ) l + z + i Takig expoetials we reach to the ifiite product for the Riema Xi-fuctio as a spectral determiat (fuctioal determiat over the Eigevalues of H) ( H + z ) ( γ + z ) z ξ ( + z ) + H E ξ γ det / det / () If we choose the positive brach f ( x) of the iverse / d f ( x) argξ i x / + the the potetial will be always positive so the dx + Eergies of the Hamiltoia iside (8) will be all positive E γ R, the all the o-trivial zeros of the Riema Zeta fuctio will be o the critical lie Re( s ), with a simple chage of variable z s we obtai det H s( s) ξ ( s) + 4 ξ ( s) s ξ () () ξ () ρ ρ det H + 4 Equatio () is the Hadamard product for the Riema Xi-fuctio i terms of the quotiet of fuctioal determiats, sice the expected value of the Hamiltoia is positive ψ H ψ ad Hermitia,with f ( x) the all the Eergies are positive E s( s) R + Riema Hypothesis should hold. 6

7 o Bohr-Sommerfeld quatizatio coditio ad the square of the Riema zeros: / d f ( x) argξ + i x / The expressio dx Bohr-Sommerfeld quatizatio coditios [7] could also be obtaied from the C pdq + a dx E f ( x) p( x) E f ( a) (3) a is the classical turig poit, N( E) is the Eigevalue staircase, the first itegral iside (3) is a lie itegral take over the closed orbit of the classical system, equatio (3) ca be uderstood as a itegral equatio for the iverse of the potetial i the form a a( E ) E df / + ( E) E V ( x) dx E x Dx f ( x) dx (4) If we take the half derivative o both sides of (4) we would get / d f ( x) argξ i x / dx + + i this case this result is completely equivalet to the oe we got by Zeta regularizatio ad by the WKB tf ( x) approximatio of the Theta fuctio dxe ΘWKB ( t) t. / d f ( x) argξ + i x / I order to evaluate the iverse of the potetial dx we would eed to evaluate arg ζ i x +, this ca be made usig the Riema-Siegel formula [] to evaluate the zeta fuctio o the critical lie ( ϑ k k ) U ( k ) iϑ ( k ) cos l Z( k) ζ + ik e + O k / 4 (5) k k The fuctios iside (5) are u( k), [ ] x is the floor fuctio ad T T T T ϑ( T ) arg Γ + i l l e 8 48T From equatio (4) the desity of states could be evaluated as / d f ( x) ρ( x) / δ ( x γ ), the desity of states or trace of dx Tr δ E f ( x) depeds o the half-derivative of the iverse of the potetial for { } 7

8 the Hamiltoia, we will prove i the ext sectio that this desity of states reproduces a distributioal versio of the Riema-Weil explicit formula o Riema Weil explicit formula as the Trace Tr { ( E f ( x) ) } δ : The ext questio is to compute the desity of states for the Hamiltoia desfied i (8), let be the property of the delta fuctio p E ( p ) + ( p + ) δ γ δ γ δ ( E γ ), if we use Shokhotsky s formula for the delta p fuctio lim I m δ ( x a) Tr δ E f ( x) ε x a + iε, the desity of states { } d arg ξ + iε + i E δ ( E γ ) δ ( p γ ) E de γ γ ζ ζ ' l Γ ' p + ip + ip + + i + ζ p ζ p p Γ 4 4 p i i δ p + δ p + Γ ' p i + ρ( E) Γ 4 4 p p (6) i Here lim I m δ x ±, this factor comes from the logarithmic ε x ± i + iε derivative of s( s ) alog the critical lie s + ip, equatio (6) is a distributioal versio of the Riema-Weil trace formula, takig formally the logarithm of the Euler product for the Riema Zeta fuctio o the critical lie l yields to Λ ip ' e ζ reg + ip ζ, usig two test fuctios h(x) ad g(x) g( x) dr cos( rx) h( r) we recover the oscillatory part of the Riema-Weil trace formula Λ g (l ). I geeral the formula (6) for the desity of states ca be obtaied by takig the Laplace trasform of the Theta fuctio Θ ( t) exp( te), i our case the WKB Theta fuctio t / d f ( x) argξ + i x / tf ( x) dxe ΘWKB ( t) with the potetial dx is equal to Θ ( t) exp( te), if we use the lim I m δ x ad s dt exp ( st ) ε x + iε the two idetities 8

9 t ( iε + E ) d Im dte ( t) ρ( E) Argξ ip ε δ ( p γ ) Θ + + (7) p dp With ε ad E p Ulike the model of Wu ad Sprug, we have cosidered also the oscillatory part of the Riema Eigevalue Staircase arg ζ i E +, which satisfy Bolte s semiclassical law, Wu ad Sprug [7] cosidered oly the smooth part T T of teh Eigevalue staircase i the limit T >> l N( T ) i order to get e a Hamiltoia whose Eergies are the positive imagiary part of the Riema Zeros, their startig poit is the Harmoic oscillator [5], but ulike the ormal quatum mechaical oscillator whose fuctioal determiat gives the Gamma s fuctio + the product take ONLY over the positive imagiary Γ( s) part of the zeros (eve if it coverges) s + γ has o meaig, also the Wu-Sprug model does t obey Weyl s law i oe dimesio maily d / N ( E) O E, i our case, the Hamiltoia (8) with the Smooth part of smooth E E the Eigevalue staircase N( E) log e, satisfies a Weyl s law with ε ( E) E d + ad the spectral determiat (quotiet) E γ is () E proportioal to the Riema xi fuctio o the critical lie ξ + i E By aalogy with the zeros of the Selberg Zeta fuctio, is better to cosider the case with the Eergies E γ, i this case the Trace of the Resolvet of the Hamiltoia ( E iε H ) + is the Riema-Weil trace for the Riema zeros. o Aalytic expressio for the potetial f ( x ) : From the expressio for the fractioal derivative of powers k λ d x Γ ( λ + ) λ k x k, we ca obtai for the iverse fuctio dx Γ λ k + / d H ( x γ ) f ( x) argξ i x / + dx γ > (8) x γ Usig the Riema-Weil formula we ca rewrite (8) as 9

10 8 dr Γ ' ir Λ l (,l ) (9) x f x + g x 8x x x r Γ x cos( ut) Here g( u l, x) dt x t, a fial questio is could the expressio / d f ( x) argξ + i x / dx be iverted to get f ( x )?,the smooth part of E E the Eigevalue staircase is give by N( E) log e, e, if we! ε x use the expressio for the logarithm log( x) as ε ad apply the half ε derivative expressio, the the followig holds ε f ( ) ε / ε / 4 e A ε x B smooth x ε ε ε x + B fsmooth ( x) 4 e A( ε ) (3) 3+ ε Γ A( ε ) 3 ad B Γ ε, the secod expressio iside (3) is Γ + the asymptotic of f ( x ) as x, for this potetial, the eergies iside (8) are smooth 4 E f N smooth ( E) W ( e ) ( ) W ( x) x (3)! The fuctio W ( x ) is the Lambert fuctio (pricipal brach), more tha i the / d potetial f ( x) argξ i x R / + we are iterested i the Theta dx fuctio Θ ( t) exp( tγ ), if we use the semiclassical Theta fuctio as a itegral over the Phase space ad itroduce the potetial give by / d f ( x) argξ i x ax + R / oe obtais, a dxe dx th ( x, p) tf ( x) t tr WKB ( t) dp dxe dxe dre f ( r) t Θ t H ( r γ ) tr t tx tγ dre e dx e ( t) exp( tγ ) γ r γ Θ γ (3)

11 I (3) we have obtaied the Heat fuctio Θ ( t) exp( tγ ), from the f ( x) / d argξ + i / x R potetiall fuctio of course to be correct we dx must take the smooth ad the oscillatory part of the Eigevalue staircase / d f ( x) / ( Nsmooth ( x) + Nosc ( x) ) otherwise the descriptio will be ot dx complete as i the Wu-Sprug potetial [7], from this Theat Kerel th ( x, p) here t> ad the Hamiltoia has bee Θ ( t) exp( tγ ) dp dpe defied i (8) usig the Zeta regularizatio method for the determiat ζ (, ( )) l det ( ( )) ζ (, ) ( ) s H s z z H z z s { } s z z Tr H z z H + (33) ζ H ( s, z( z )) s, the zeros of the determiat + z( z ) + γ 4 det H z( z) with H a Hermitia operator are the zeros of ξ ( z) o Numerical evaluatio of the fuctioal determiat : We eed to evaluate the half-derivative iside the iverse of the potetial / d f ( x) argξ i x / +, to do so we ca use the Gruwald-Letikov dx formula [3] with a step ε. ad q q g x d g x Γ q + + ε Γ ( + ) Γ( + ) / q / q dx ε q g ( x ( q ) ε ) (34) For the case of the fuctioal determiat of our Hamiltoia operator with the / d potetial f ( x) argξ i x / + defied as dx d + + f ( x) λ y( x, λ) y(, λ) y( L, λ) dx 4 L (35) λ s( s), to evaluate the fuctioal determiat by the Gelfad-Yaglom method [8] we eed to solve the iitial value problem

12 d dy(, λ) + + f ( x) λ y( x, λ) y(, λ) dx 4 dx (36) Ufortuately exact solutios to (36) ca ot be foud, i the WKB approximatio (36) has the solutio x x y( x, λ ) C exp ( t) dt C exp ( t) dt / + Π + Π Π( x) Π ( x) f ( x) + λ (37) 4 The costats C ± are chose so (37) solves the iitial value problem (36). The Gelfad-Yaglom theorem [8] tells us that the fuctioal determiat is related to the solutio of the iitial value problem (36) i the form det H s( s) y( L, λ) + 4 ξ ( s) s lim L y( L,) λ s( s) (38) ξ () ρ ρ det H + 4 The mai advatage of the Gelfad-Yaglom method, is that we do ot eed to evaluate ay sigle eigevalue i order to obtai the fuctioa determiat det H + λ, ufortuately this method is oly valid for ordiary differetial 4 equatios TABLE : compariso betwee the Riema Zeros (square) from the tables of Odlyzko ad the Numerical values of the eergies for our Hamiltoia operator (8) with / d f ( x) argξ i x / +, to obtai umerically the potetial we have used formula dx (34) to evaluate the fractioal derivative ad the Riema-Siegel formula (5) to evaluate S( T ) argζ + it Zeros (square) Eergies

13 Refereces [] Abramowitz, M. ad Stegu, I. A. (Eds.). "Riema Zeta Fuctio ad Other Sums of Reciprocal Powers." 3. i Hadbook of Mathematical Fuctios. New York: Dover, pp , 97. [] Apostol Tom Itroductio to Aalytic Number theory ED: Spriguer-Verlag, (976) [3] Berry M. Keatig J.P The Riema Zeros ad Eigevalue Asymptotics Siam Review archive Vol. 4 Issue, Jue 999 pages: [4] Bolte J: "Semiclassical trace formulae ad eigevalue statistics i quatum chaos", Ope Sys. & Iformatio Dy.6 (999) 67-6, avaliable at: [5] Cartier P. ad Voros A. Ue ouvelle iterpretatio de la formule des traces de Selberg. I P. Cartier, L. Illusie, N. M. Katz, G. Laumo, Y. Mai, ad K. A. Ribet, editors, The Grothedieck Festschrift (vol. II), volume 87 of Progress i Mathematics, pages 67. Birkh auser, 99. [6] Elizalde E., Oditsov S.,Romeo A., Bytseko A., Zerbii S Zeta Regularizatio Techiques With Applicatios ISBN-: Ed: World Scietific pub (994) [7] Griffiths, David J. (4). Itroductio to Quatum Mechaics Pretice Hall. ISBN [8] Grosche Path Itegrals, Hyperbolic Spaces ad Selberg trace formulae (996) ISBN-: 9843 [9] Gutzwiller, M. Chaos i Classical ad Quatum Mechaics, (99) Spriger- Verlag, New York ISBN [] Hejhal D. The Selberg Trace formula ad the Riema Zeta fuctio Duke Mathematical Joural 43 (976) [] Hurt N. Quatum Chaos ad Mesoscopic Systems: Mathematical Methods i the Quatum Sigatures of Chaos () ISBN-3: [] Marklof J. Selberg Trace formula: A itroductio PDF avaliable at [3] Nishimoto, K. Fractioal Calculus. New Have, CT: Uiversity of New Have Press,

14 [4] Odlyzko A. Tables of Zeros of Riema Zeta fuctios,webpage [5] Selberg, Atle (956), "Harmoic aalysis ad discotiuous groups i weakly symmetric Riemaia spaces with applicatios to Dirichlet series", J. Idia Math. Soc. (N.S.) : 47 87, MR885 [6] Voros A. Exercises i exact quatizatio e-prit avaliable at [7] Wu H. ad D. W. L. Sprug Riema zeros ad a fractal potetial Phys. Rev. E 48, (993) [8] Yaglom, A.M Gelfad I.M Itegratio i fuctioal spaces ad its applicatios i quatum physics, J. Math. Phys.,