Concerning Riemann Hypothesis

Concerning Riemann Hyp othesis R. Ac harya ∗ Department of Ph ysics & Astr o nom y Arizona State Universit y T emp e, AZ 8 5287-1 504 Marc h 1 7, 2009 Abstract W e present a qu an tum mec hanical mod el wh ic h establishes the v eracit y of the Rie mann h yp othesis that the non-trivia l zeros of the Riemann zeta-function lie on the critical line of ζ ( s ). ∗ I dedicate this note to my teac her George Sudarsha n and to the memor y of Sriniv asa Ramanujan (22 Decem be r 1887 – 26 April 1920), “The Man Who K new ‘Infinit y’.” 1 W e b egin b y recalling the all-to o- familiar lore that the R iemann h ypot h- esis has b een the Holy Grail of mathematics a nd phys ics for more than a cen tury [1]. It asserts that all the zeros of ξ ( s ) ha v e σ = 1 2 , where s = σ ± it n , n = 1 , 2 , 3 . . . ∞ . It is b eliev e d all zeros o f ξ ( s ) a re simple. T he function ζ ( s ) is related to the Riemann ξ ( s ) function via the defining relation [1], ξ ( s ) = 1 2 s ( s − 1) π − S 2 Γ  s 2  ζ ( s ) (1) so that ξ ( s ) is an en tire function, where ζ ( s ) = ∞ X n =1 n − s , s = σ + it, σ > 1 (2) ζ ( s ) is holomorphic for σ > 1 and can ha v e No zeros for σ > 1. Since 1 / Γ( z ) is en tire, the f unction Γ  s 2  is non-v anishing, it is clear that ξ ( s ) also has no zeros in σ > 1: the zeros of ξ ( s ) a re confined to the “ critical strip” 0 6 σ 6 1. Moreo v er, if ρ is a zero of ξ ( s ) , then so is 1 − ρ and since ξ ( s ) = ξ ( s ), one deduces that ρ and 1 − ρ are also zeros. Th us the Riemann zeros are sym metrically a rranged abo ut the real axis and also ab out the “critical line” giv en b y σ = 1 2 . The Riemann Hypot hesis, then, asserts that ALL zeros o f ξ ( s ) hav e Re s = σ = 1 2 . W e conclude this in tro ductory , w ell- kno wn remarks with the assertion that ev ery en tire function f ( z ) of order one and “ infinite t yp e ” (whic h guaran tees the existence of infinitely man y Non-zero zeros can b e r epre- sen ted b y the Hadamard factorization, to wit [2], f ( z ) = z m e A e B Z ∞ Y n =1  1 − z z n  exp  z z n  (3) where ‘ m ’ is the multiplicit y of the zeros (so that m = 0, for simple zero). Finally , ξ ( s ) = ξ ( 1 − s ) is indeed an en tire function of order one and infinite type and it has No zeros either for σ > 1 or σ < 0. W e inv ok e the w ell-kno wn results of Lax-Phillips [3] and F addee v-P a vlo v [4] scattering theory of automorphic functions in the P o incare’ upper-half plane, z = x + iy , y > 0, −∞ < x < ∞ , whic h w as motiv ated by Gelfand’s [5] observ ation of the analogy b et w een t he Eisenstein functions [6] and the scattering matrix, S ( λ ). Recen tly , Y oic hi Uetake [7] has undertak en a de- tailed study of Lax-Phillips and F a ddeev-P avlo v analysis, b y resorting t o the tec hnique of Eisenste in transform. 2 W e recast this result b y sp ecializing to the case where incoming and out- going subspaces are necessarily orthogonal , D − ⊥ D + and by restricting to the case of Laplace-Beltrami op erator with constan t ‘ x ’ and iden tifying the r esulting w a v e equation, as a one-dimens ional Sc hro dinger equation, at zero energy , with a re pulsiv e, in v erse-sq uare p oten tial, V ( y ) = λy − 2 , λ > 0, y > 0 and an infinite barrier at the origin , V ( y ) = ∞ , y 6 0. W e obtain zero- energy Jost [8, 9] function F + ( k 2 = 0): S − 1 ( k 2 = 0) = F + ( k 2 = 0) = ξ ( − 2 s ) ξ (2 s ) (4) where ξ ( s ) is Riemann’s ξ f unction [1], Eq.(1) and we ha ve shifted the v ariable ‘ s ’ b y 1 2 , following the con ven tion of Uetak e [7]. Since all zeros of Jost function F + ( s ) lie on the critical line, Rs = − 1 4 , w e conclude that the Riemann h yp o thesis is v alid. As an in tro duction to set the not ation, w e b egin by presen ting the familia r case of the Euclidean pla ne, R 2 = { ( x, y ) : x, y ∈ R } (5) The group G = R 2 acts on itself a s translations, and it mak es R 2 a homogeneous space. The Euclidean plane is iden t ified b y the metric dL 2 = dx 2 + dy 2 (6) with zero curv ature ( K = 0) and the Laplace-Beltrami op erator asso ciated with this metric is giv en b y D = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 (7) Clearly , the exp onen tial functions ϕ ( x, y ) = exp[2 π i ( ux + v y )] , ( u, v ) ∈ R 2 (8) are eigenfunctions of D : ( D + λ ) ϕ = 0 , λ = λ ( ϕ ) = 4 π 2 ( u 2 + v 2 ) (9) The upp er-half h yp erb olic half-plane [3] ( called the Poincare’ plane) is iden tified b y 3 | H | = { z = x + iy : x ∈ R , y ∈ R + } (10) | H | is a Riemann manif o ld with the metric dL 2 = y − 2 ( dx 2 + dy 2 ) (11) It represen ts a mo del of non-Euclidean g eometry , where the role of non- Euclidean motion is t a k en b y the group G of fractional linear transformations, W → aw + b cw + d (12) with ab − bc = 1 , a, b, c, d real (13) The matrix  a b c d  and its negativ e furnish the same transformation. The Riemannian metric, Eq.(11) is in v ariant under this group of motions. The Laplace-Beltrami op erator is giv en by y 2 ∆ = ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) (14) A discrete subgroup of in terest is the mo dular group consisting of trans- formations with in teger a , b , c , d . A fundamen tal domain F for a discrete subgroup Γ is a sub domain of the Poincare’ pla ne suc h that ev ery p oin t of Π can b e carried in to a p oint of the closure F of F b y a transformatio n in Γ and no p oin t of F is carried in to another p oin t of F b y suc h a transformatio n. F can b e rega rded as a manifold where tho se b oundary p oin ts which can b e mapp ed in to eac h other b y a γ in Γ are iden tified. Then, a function f defined o n Π is called auto morphic with resp ect to a discrete subgroup Γ if f ( γ w ) = f ( w ) (15) for all γ in Γ. By virtue o f Eq.(15), an automorphic function is completely determin ed b y its v alues on F . The Laplace-Beltrami op erator , Eq.(14) maps automor- phic f unctions in to automorphic functions. In regular co ordinates, z = z + iy , if w e require f ( z ) to be a function of y only , i.e., constan t in x , w e arriv e at 4 ∂ 2 f ( y ) ∂ y 2 + λ 0 f ( y ) y 2 = 0 , y > 0 (16) with t w o inde p enden t solutions [6 ], 1 2 ( y s + y 1 − s ) and 1 2 s − 1 ( y s − y 1 − s ) (17) where λ 0 = s (1 − s ) . (18) F o r s = 1 2 ( λ 0 = 1 4 ), the ab o v e eigenfunctions b ecome y 1 2 and y 1 2 log y resp ectiv ely . W e no w mak e an imp ortan t observ ation whic h is crucial , i.e., we can view Eq.(16) a s a Sc hro dinger equation at zero energy , for an in v erse-square p oten tial, i.e., ∂ 2 Ψ( y ) ∂ y 2 + ( k 2 − V ( y ))Ψ( y ) = 0 (19) where k 2 = 0  2 m ~ 2 = 1  and V ( y ) = s ( s − 1) y 2 (20) with the all-imp ortan t constrain t that V ( y ) = s ( s − 1) y 2 , y > 0 (21) [ s > 1 or s < 0] i.e. R epulsiv e! and V ( y ) = ∞ , y 6 0 . (22) In other words, the restriction on v ariable ‘ y ’ in the P oincare’ upp er-half plane, Eq.(10) requires that y > 0 and Eq.(22) ensures that this requiremen t is satisfied b y placing a n “infinite barrier” at y = 0, so that Ψ( y ) ≡ 0, in the “left-half” y -a xis. W e no w follow Uetak e’s analysis closely [7].The Eisenstein series of t w o v ariables E ( z , s ) on a fundamental domain is b y definition, is real analytic for z = x + iy ∈ | H | and E ( z , s ) is meromorphic in s in the complex plane C . It is regular f or eac h z ∈ | H | , with respect to ‘ s ’ in Rs > 0, except at s = 1 2 . In fact, E ( z , 1 2 + s ) is an aut o morphic function on | H | : 5 E  γ ( z ) , 1 2 + s  = E  z , 1 2 + s  ∀ γ ∈ S L 2 ( z ) . (23) In Eq.(23), following Uetak e, w e ha v e shifted the v ar ia ble ‘ s ’ b y 1 2 and γ ( z ) = z + n for some n ∈ Z . Th us, E ( z , 1 2 + s ) can b e view ed as a function on | H | (upp er-half P oincare’ plane.) F urthemore, the Eisenstein series is a (non L 2 − ) eigenfunction of the non-Euclidean Laplacian, − y 2  ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2  on | H | , i.e.,  − y 2  ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2  − 1 4  E ( z , 1 2 + it ) = t 2 E ( z , 1 2 + it ) (24) for all t ∈ C . In tegrating Eq .(24 ), w.r.t x yields Eq.(19) and Eq.(20) where s → s + 1 2 [3]. Eq.(24) is an eigenfunction pro p ert y of the Eisenstein series. Also, the Eisenstein series satisfies the functional equation, E  z , 1 2 + s  = S ( s ) E  z , 1 2 − s  (25) where S ( s ) is the scattering “matrix.” Eq.(24) and Eq.(25) are prov en, for instance, in Y. Motohashi’s text on the sp ectral theory of the Riemann Zeta- function [10]. When E  z , 1 2 + s  is expanded in the F ourier series of exp( inz ), the zero F o ur ier co efficien t tak es the form, y 1 2 + s + S ( s ) y 1 2 − s (26) where the scattering matrix S ( s ) has the follo wing f orm, for orthogonal incoming and outgoing subspaces D − and D + [7]: S ( s ) = ξ (2 s ) ξ ( − 2 s ) (27) where Riemann’s ξ ( s ) function is related to THE Riemann function ζ ( s ) via Eq.(1) [2]: ξ ( s ) = 1 2 π − S 2 s ( s − 1)Γ  s 2  ζ ( s ) (1) Imp ortan tly , S ( s ) has p o les in − 1 2 < Rs < 0 a nd the Riemann h ypo thesis is the a ssertion that all p oles of S ( s ) lie on Rs = − 1 4 . 6 W e mov e on next to mak e contact with the Jost functions [8] F + − ( k ) and the S matrix. It is well-kno wn that the Jost function is iden tified via the defining relation, S ( k ) = F − ( k ) F + ( k ) (28) where F + ( k ) is holomorphic in the upp er-half complex k -plane and F − ( k ) is holomorphic in the lo w er-half k -plane, where k is the momen tum. Th us, F + ( k ) = S − 1 ( k ) F − ( k ) , I mk = 0 , −∞ < k < ∞ (29) and det S ( k ) 6 = 0 , I mk = 0 (30) The “solution” to the b oundary v alue problem w as form ulated by Krutov, Mura vy ev and T roitsky in 1 9 96 [11]. It reads: F + − ( k ) = Π + − ( k ) exp  1 2 π i Z ∞ −∞ ln( S − 1 ( k ) π 2 − ( k ) k ′ − k ∓ io dk ′  (31) where Π + − ( k ) = m ′ Y j =1 k ∓ ik j k ± ik j , k j > 0 , j = 1 , 2 . . . m ′ ( m ′ < ∞ ) (32) and Π + − ( k ) ≡ 1 , m ′ = 0 (33) where m ′ is the num b er o f b ound states. It is, of course, well known that 1 k ′ − k ∓ io = P 1 k ′ − k ± iπ δ ( k ′ − k ) (34) Pro ceeding further, w e set k = 0 (z ero energy). The simplification is immediate. One finds that the zero energy Jost functions are given by: S − 1 ( s ) = F + ( s ) , F − ( s ) = 1 (35) 7 Th us, w e arriv e at t he stated result, Eq.(14): S − 1 ( s ) = F + ( s ) = ξ ( − 2 s ) ξ (2 s ) (4) The Riemann h yp othesis is the assertion that all p oles of S ( s ) lie o n Rs = − 1 4 . F rom Eq.(4 ), we see that this requires that all zeros of the Jost function, F + ( s ) must lie on R s = − 1 4 . It is w ell-kno wn from the the theory o f Jost functions, summarize d by Kh uri [1 2] that the s -wa ve Jost function for a p otential is an en t ir e func- tion of the “coupling constan t” λ , with an infinite n um b er of zeros extending t o infinit y . F or a repulsiv e p oten tial V ( y ) and at zero energy , these zeros of λ n will all b e real and negative , λ n (0) < 0. By c hanging v ariables to ‘ s ’ ( in Uetak e’s c on ven tion ), with λ = s ( s − 1 ), it follows that as a function of ‘ s ’, the Jost function F + ( s ) ha s only zeros on t he line S n = − 1 4 ± it . (Again, in Uetake’s [7] conv en tion for ‘ s ’ !) This is automat ic, in view of Eq.(21)! At this p oin t, t he discussion in Kh uri detailed in his Eq.(1.1), Eq.(1.2) and Eq.(1.3) demonstrates the j us- tification (require d). While Khuri’s analysis [12] dealt with exp onen tia lly decreasing p oten tials for x → ∞ , the presen t situation of in v erse-square p o- ten tial can b e dealt with b y taking a sligh t detour in a rriving at the analogue of K h uri’s Eq.(1.2), i.e., it is easily demonstrated that [ k = iτ ]: I m [ λ n ( iτ )] Z ∞ 0 | f ( λ n ( iτ ); iτ , y | 2 dy = 0 (36) T o see this, one writes do wn the differen tial equation for the Jost solu- tion , f ( λ ; k ; y ) [whic h is iden tical to the s -wa ve Jo st function in 3 dimen- sions!] and its complex conjugate partner, then, one can p erfo rm the “canon- ical” op eration of m ultiplying the equation for f ( y ) by f ∗ ( y ) and like-wis e for f ∗ ( y ) equation b y f ( y ), doing the subtraction and multiply through b y V ( y ) whic h is real!] Finally , expressing the t erms in v olving f f ∗ ′′ − f ′′ f ∗ as deriv a t iv e of the wronskian whic h is indep enden t of y (!), one finally ends up with Eq.(36). It is an elemen tary exercise to p erform the in tegral in Eq.(36) b y no t ing that the Jost solution has the form: f ( k , y ) = r π k y 2 e i ( π 2 ν + π 4 ) H (1) ν ( k y ) (37) 8 where H (1) ν ( k y ) is Hankel function of I kind and it has the required asymp- totic b ehavior, i.e., f ( k , y ) − → y → ∞ e ik y (38) One notes tha t [1 3] Z ∞ 0 y K 2 ν ( iτ ) ( y ) d y = 1 8 π ν ( iτ ) sin π ν ( iτ ) , ν ( iτ ) = r λ ( iτ ) + 1 4 (39) where ν ( iτ ) 6 = 1 , 2 , 3 , . . . ∞ . [ I mλ n ( iτ )] 1 τ 2 π Z ∞ 0 y K 2 ν ( iτ ) ( y ) d y = 0 (40) Since the in tegral in Eq.(40) is finite and non- v anishing [k eeping ν 6 = 1 , 2 , 3 . . . ∞ ], one ends up with (following Khuri), t a king the limit τ → 0 and get I mλ n (0) = 0 (41) The zero energy , coupling constan t sp ectrum, λ n (0) m ust lie on the neg- ativ e real axis for V > 0 ( Repulsiv e pot ential). The rationale b ehind the idea of “getting rid of” the p o t ential, V ( y ) = λ y 2 in Eq.(36) is to ensure that the resulting equation (Eq.(36)) is no w finite and one can then con tin ue on to conclude that the zero energy , c oupling c onstan t λ n (0) m ust lie on the negativ e real line, for V > 0. In conclusion, the deriv ation o f Jost function, F + ( s ) = ξ ( − 2 s ) ξ (2 s ) where ξ ( s ) is Riemann’s ξ function and the established k ey assertion that ALL zeros of F + ( s ) must lie on the critical line, Rs = − 1 4 , leads us to the conclusion that the Riemann’s h yp othesis is v a lid [14 ]. Ac kno wledgemen t: I am indebted to Irina Long for her kind assistance. References : [1] “The Riemann Hyp othesis: A Resource for the Afficinado and Virtuoso Alik e,” Borw ein, Choi, Ro oney and W eirathm ueller ( Editor s). 9 [2] “The Theory of the Riemann Zeta-F unction,” Second Edition, E. C. Titc hmar sh, revis ed by D. R. Heath- Bro wn, Clarendon Press, Oxford [1986]. [3] P . D . Lax and R. S. Phillips, “Scattering Theory of Automorphic F unc- tions,” Ann. of Math. Studies, no. 87, Princeton Univ. Pres s, Princeton, New Jersey (1976). [4] L. D. F addeev a nd B. S. P a vlo v, “Scattering theory and automorphic function,” J. Soviet. Math 3 (1975), 522– 5 48. [5] I. M. G elfand, “Automorphic F unctions and the Theory of R epresen ta- tions,” Pro c. In t. Cong. of Math., Sto c kholm (1962), 74–85. [6] See for instance, H. Iw a niec, “Sp ectral metho ds of auto morphic fo rms,” Second Edition, Graduate Studies in Mathematics, v. 53. [7] Y oichi Uetak e, “Lax-Phillips scattering for automor phic functions based on Eisenstein transform,” Inte gr al Equations and Op er ator T he ory 60 (2008), 271–28 8 . [8] R. Jost and A. Pais, Phys. R ev. 82 (1 951), 8 40. [9] K. Chadan and R. C. Sabat ier, “In v erse Problems in Quantum Scatter- ing Theory ,” Second Edition, Springer-V erlag, N.Y. 1989. [10] Y. Motohashi, “Sp ectral Theory of the R iemann Zeta - F unction,” Cam- bridge T racts in Math. 127 , Cam bridge Univ ersit y Press (1997) . [11] A. F. Krutov, D. I. Mura vy ev, and V. E. T roitsky , “A factorization of the S -matrix into Jost matrices,” J. Math. Phys. 38 (1997), 2880. [12] N. N. K huri, “In v erse scattering, the coupling constant sp ectrum and the Riemann Hyp othesis,” a rXiv: hep-th/01 1 1067, v. 1 , No v em b er 2001. [13] I. S. Gg radsh teyn and I. M. Ryzhik, “T ables of In tegrals, Series and Pro ducts,” Corrected and Enlarg ed Edition, Academic Press (1980), page 693. 10 [14] There a re inn umerable a t tempts at proving the Riemann h yp othesis and to o many (“ ∞ ”!) to justify here! The imp ortan t work of M. V. Berry , “Lecture Notes in Ph ysics” 262 , Springer-V erlag, 19 86 and t he profound w ork b y S. Okub o, “Lorentz Inv arian t Hamiltonia n and Riemann Hy- p othesis,” J. Phys. A: Math. Gen. 31 (1998) , 1 0 49, is w orthy of men t ion, b esides Kh uri’s w ork. 11