Closed-Form Solution for the Nontrivial Zeros of the Riemann Zeta Function

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1 Closed-Form Soluion for he Nonrivial Zeros of he Riemann Zea Funcion Frederick Ira Moxley III (Daed: April 8, 27) Recenly i was conjecured ha if he Bender-Brody-Müller (BBM) Hamilonian can be shown o be self-adjoin, hen he Riemann hypohesis holds rue. Herein we discuss he domain and eigenvalues of he Bender-Brody-Müller conjecure. I. INTRODUCTION I was recenly shown in [] ha he eigenvalues of a Bender-Brody-Müller (BBM) Hamilonian operaor correspond o he nonrivial zeroes of he Riemann zea funcion [2]. Alhough he BBM Hamilonian is pseudo-hermiian, i is consisen wih he Berry-Keaing conjecure [3, 4]. The eigenvalues of he BBM Hamilonian are conjecured o be he imaginary pars of he nonrivial zeroes of he zea funcion ζ(z) = k z k= = Γ(z) z d. () exp() The idea ha he imaginary pars of he zeroes of Eq. () are given by a self-adjoin operaor was conjecured by Hilber and Pólya [5]. Formally, Hilber and Pólya conjecured ha if he eigenfuncions of a self-adjoin operaor saisfy he boundary condiions ψ n () = n, hen he eigenvalues are he nonrivial zeroes of Eq. (). The BBM Hamilonian also saisfies he Berry-Keaing conjecure, which saes ha when ˆx and ˆp commue, he Hamilonian reduces o he classical H = 2xp. Remark. If here are nonrivial roos of Eq. () for which R(z) /2, he corresponding eigenvalues and eigensaes are degenerae []. II. STATEMENT OF PROBLEM Theorem. The eigenvalues of he Hamilonian A. Bender-Brody-Müller Hamilonian Ĥ = are real, where ˆp = i x, =, and ˆx = x. e iˆp (ˆxˆp ˆpˆx)( e iˆp ) (2) Corollary.. [] Soluions o he equaion Ĥψ = Eψ are given by he Hurwiz zea funcion ψ z (x) = ζ(z, x ) = (x n) z (3) on he posiive half line x R wih eigenvalues i(2z ), and z C, for he boundary condiion ψ z () =. Moreover, R(z) >, and R(x ) >. As ψ z () is he Riemann zea funcion, i.e., Eq. (), his implies ha z belongs o he discree se of zeros of he Riemann zea funcion. Proof. Le ψ z (x) be an eigenfuncion of Eq. (2) wih an eigenvalue λ = i(2z ): Ĥψ z (x) = λψ z (x). (4) Then we have he relaion e iˆp (ˆxˆp ˆpˆx)( e iˆp )ψ z (x) = λψ z (x). (5)

2 2 Leing ϕ z (x) = [ exp( x )]ψ z (x), = ˆ ψ z (x), (6) where ˆ ψ z (x) = ψ z (x) ψ z (x ), and ˆ exp( x ), (7) is a shif operaor. Upon insering Eq. (6) ino Eq. (5) wih ˆp = i x, =, and ˆx = x, we obain Then we have [ ix x i x x]ϕ z (x) = λϕ z (x). (8) (x x ϕ z (x)) ϕ z (x)dx R ( x xϕ z (x)) ϕ z (x)dx = iλ R ϕ z(x)ϕ z (x)dx. R (9) Now we inegrae he firs erm on he LHS of Eq. (9) by pars o obain xϕ z (x) x ϕ z(x)dx = ϕ z(x)ϕ z (x)dx ϕ z(x)x d R R R dx (ϕ z(x))dx, () and he second erm on he LHS of Eq. (9) by pars o obain xϕ z (x) x ϕ z (x)dx = ϕ z (x)ϕ z(x)dx ϕ z (x)x d R R R dx (ϕ z(x))dx. () Upon subsiuing Eqs. () and () ino Eq. (9), we obain ϕ z(x)x d R dx (ϕ z(x))dx ϕ z (x)x d R dx (ϕ z(x))dx = (iλ 2)N, (2) where Nex, we spli ϕ z (x) ino real and imaginary componens, such ha N = ϕ z(x)ϕ z (x)dx. R (3) ϕ = R(ϕ z (x)) ii(ϕ z (x)), (4) and subsiue Eq. (4) ino Eq. (2) such ha R(ϕ z (x))x d R dx R(ϕ z(x))dx I(ϕ z (x))x d R dx I(ϕ z(x))dx N = iλ N. (5) 2 Upon seing λ = i(2z ), Eq. (5) can be wrien R(ϕ z (x))x d R dx R(ϕ z(x))dx I(ϕ z (x))x d R dx I(ϕ z(x))dx N = 2z N. (6) 2 I can be seen ha all erms on he LHS of Eq. (5) are real, hereby verifying Theorem. Q.E.D. Remark. If he Riemann hypohesis is correc [2], he he eigenvalues of Eq. (2) are degenerae []. Lemma.. Under he boundary condiion ψ() =, he n h eigensae of Eq. (2) is Eq. (3), and he nonrivial zeros of he Riemann zea funcion are given by z n = R(ϕ n (x))x d N R dx R(ϕ n(x))dx I(ϕ n (x))x d N R dx I(ϕ n(x))dx 3 2. (7)

3 3 Proof. Given ha ψ n (x) = ˆ ψ n (x) = ψ n (x) ψ n (x ) = (x n) z (x n) z, (8) he second erm on he RHS of Eq. (7) goes o zero, as I(ϕ n (x)) =. Hence, we are lef wih z n = ϕ n (x)x d N R dx ϕ n(x)dx 3 2. (9) Moreover, i can be seen ha Muliplying Eq. (2) by ϕ n (x), we obain x d dx (ϕ n(x)) = x d dx ψ n(x) x d dx ψ n(x ) = x d dx (x n) z x d dx (x n) z = xzζ(z, x ) xzζ(z, x). (2) ϕ n (x)xzζ(z, x ) ϕ n (x)xzζ(z, x) = ϕ n (x)[xzζ(z, x ) xzζ(z, x)] = ζ(z, x )xzζ(z, x ) ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x ) ζ(z, x)xzζ(z, x). (2) From he RHS of Eq. (2), i can be seen ha ζ(z, x )xzζ(z, x )dx = R ζ(z, x)xzζ(z, x)dx = R (n x ) 2z (n 2xz ) 2(2z ) (n x) 2z (n 2xz) 2z(2z ) cons (22) cons (23) and ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx R ((n x) z (n x ) z ((n x) 2 F (, 2z, z, n x ) n 2xz)) = cons, (24) (2z ) where he hypergeomeric funcion is () n ( 2z) n (n x ) n 2F (, 2z, z, n x ) =. (25) ( z) n n!

4 4 Since N = ϕ n(x)ϕ n (x)dx R = [ψ n (x) ψ n (x )] 2 dx R = [ψn(x) 2 2ψ n (x )ψ n (x) ψn(x 2 )]dx R = [(n x ) 2z 2(n x) z (n x ) z (n x) 2z ]dx R ζ(2z, x) 2( n x) z (n x) z (n x ) z 2F ( z, z, 2 z, n x ) = ( 2z) z (n x ) 2z, R(z) >, (26) ( 2z) wih he hypergeomeric funcion Eq. (9) can be rewrien 2F ( z, z, 2 z, n x ) = ( z) n (z) n (n x ) n, (27) (2 z) n n! [ ( 2z) z n = ζ(2z, x) z 2( n x) z (n x) z (n x ) z 2F ( z, z, 2 z, n x ) ( 2z) ][ (n x ) 2z (n 2xz ) (n x) 2z (n 2xz) (n x ) 2z 2(2z ) 2z(2z ) ((n x) z (n x ) z ((n x) 2 F (, 2z, z, n x ) n 2xz)) (2z ) for R(z) >, and R(x ) >. Upon imposing he boundary condiion Eq. (28) are he nonrivial zeros of Eq. (). ψ n () = n z n= = Γ(z) ] 3 2, (28) z exp(), (29) B. Convergence For breviy, le us examine he convergence of he inegral represenaion of he discree nonrivial zeros of he Riemann zea funcion on he posiive half line x R, z C, R(z) >, and R(x ) >. From Eq. (9), he inegral represenaion of he discree nonrivial zeros of he Riemann zea funcion are given by z n = ζ(z, x )xzζ(z, x )dx N R ζ(z, x)xzζ(z, x)dx N R ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx 3 N R 2, (3)

5 5 where N = [(n x ) 2z 2(n x) z (n x ) z (n x) 2z ]dx. R (3) Lemma.2. From he firs erm on he RHS of Eq. (3), if exiss for every number, hen ζ(z, x )xzζ(z, x )dx (32) ζ(z, x )xzζ(z, x )dx provided his limi exiss as a finie number. Proof. ζ(z, x )xzζ(z, x )dx, (33) From L Hospial s Rule, we have ζ(z, x )xzζ(z, x )dx = ((n ) 2z ((n )( (n) )2z n 2z )) (2z(2z )) (34) lim ((n ) 2z ((n )( (n) )2z n 2z )) (2z(2z )) ((n ) 2z ((n )( (n) )2z n 2z )) (n ) 2z (2z(2z )) (n ) 2z 2z(n ) 4z (n( (n) )2z ( (n) )2z n 4z ) 4( 2z)z 2 (n ) 2z. (35) Upon evaluaing Eq. (35) wih a series expansion a =, we obain ( n ( n)( /( n)) 2z 4z) lim (2( n ) 2z z( 2z)) = ((n ) 2z ((n )( (n) )2z n ( 4z) )). (36) (2z(2z )) Hence, i can be seen ha he firs erm on he RHS of Eq. (3) is convergen, given ha he limi seen in Eq. (35) exiss as a finie number as seen in Eq. (36). Lemma.3. From he second erm on he RHS of Eq. (3), if exiss for every number, hen ζ(z, x)xzζ(z, x)dx (37) provided his limi exiss as a finie number. Proof. ζ(z, x)xzζ(z, x)dx ζ(z, x)xzζ(z, x)dx, (38) ζ(z, x)xzζ(z, x)dx = ((n ) 2z ( n( (n) 2z n n 2z)) (2(2z )) (39)

6 6 From L Hospial s Rule, we have ((n ) 2z ( n( (n) 2z n n 2z)) lim (2(2z )) ((n ) 2z ( n( (n) 2z n n 2z)) (n ) 2z = lim (2(2z )) (n ) 2z (n ) 4z ( n( (n) n = lim )2z n 2z) 2(2z )(n ) 2z (4) Upon evaluaing Eq. (4) wih a series expansion a =, we obain ((n ) 2z ( n((n )/n) 2z n 2z)) lim ((n ) 2z ( n((n )/n) 2z n 2z)) = ((n ) 2z ( n( (n) n )2z n 2z)). (4) (2(2z )) Hence, i can be seen ha he second erm on he RHS of Eq. (3) is convergen, given ha he limi seen in Eq. (4) exiss as a finie number as seen in Eq. (4). Lemma.4. From he hird erm on he RHS of Eq. (3), if exiss for every number, hen provided his limi exiss as a finie number. Proof. From he RHS of Eq. (24) i can be seen ha ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx (42) ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx, (43) ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx = ((n ) z (n ) z ((n ) 2 F (, 2z, z, n ) n 2z)) (2z ) ((n) z (n ) z ((n) 2 F (, 2z, z, n ) n)). (44) (2z ) Since he second erm on he RHS of Eq. (44) is independen of, we are only concerned wih he limi of he firs erm on he RHS of Eq. (44). As such, ((n ) 2 F (, 2z, z, n ) n 2z) lim (n ) z (n ) z (2z ) ((n ) 2 F (, 2z, z, n ) n 2z) (n ) z (n ) z 2F (, 2z, z, n ) (2z ) 2F (, 2z, z, n ) 2F (, 2z, z, n )(n 2 F (, 2z, z, n ) 2 F (, 2z, z, n ) n 2z) 2z(n ) z (n ) z 2F (, 2z, z, n ) (n ) z (n ) z 2F (, 2z, z, n ) (45)

7 7 C. Domain of he Bender-Brody-Müller Hamilonian For he BBM Hamilonian operaor as given by Eq. (2), he Hilber space is H = L 2 (R, dx). Moreover, ˆp and ˆx are self-adjoin operaors ha ac in H. In order o sudy he domain of he BBM Hamilonian operaor, we firs inroduce an auxiliary operaor Ô, such ha Ô = ˆpˆp ˆxˆx, (46) where ˆpˆp = 2, and ˆxˆx = x 2. The se of finie linear combinaions of Hermie funcions is a core of Ô, and herefore he Schwarz space S is also a core of Ô. Lemma.5. [6] If ϕ is in D(Ô), hen ˆpˆpϕ 2 ˆxˆxϕ 2 Ôϕ 2 c ϕ 2. (47) Proof. [6] We esimae ϕ for a core of Ô via a double commuaor o make a double commuaor esimae [7], Therefore, in Eq. (47) c = 2n. Afer rewriing Eq. (8) as ÔÔ = ˆpˆpˆpˆp ˆxˆxˆxˆx ˆpˆpˆxˆx ˆxˆxˆpˆp n [ ] = ˆpˆpˆpˆp ˆxˆxˆxˆx 2 ˆx i ˆpˆpˆx i [ˆx i, [ˆx i, ˆpˆp]] i= ˆpˆpˆpˆp ˆxˆxˆxˆx 2n, (48) [x x x x]ϕ = ( 2z)ϕ, (49) hen ˆpˆp = x x and f(ˆx) = x x are self-adjoin operaors acing in H = L 2 (R, dx). Seing defined on If f(ˆx) is local in H, hen Eq. (5) is dense and Hermiian. Ĥ = ˆpˆp f(ˆx), (5) ˆpˆp : X f(ˆx) : Y. (5) Theorem 2. The BBM Hamilonian operaor in Eq. (2) is essenially self-adjoin, given ha f(ˆx) a ˆx b. The BBM Hamilonian operaor in Eq. (2) is real-valued on he posiive half line R, afer being reduced o Eq. (49). From f(ˆx) a ˆx b we have Le us examine he uniqueness. f(ˆx) ˆxˆx b ˆx 2 cˆxˆx d. (52) Proof. As shown in [6], if Ĥ is Hermiian, and Ô is a posiive self-adjoin operaor, hen C is a core of Ô such ha C D(Ĥ). As such, (ˆpˆp f(ˆx))ϕ 2 a (ˆpˆp ˆxˆx)ϕ 2 b ϕ 2, (53) where ϕ S. Since ( ˆxˆx)ϕ L 2, f(ˆx)ϕ L 2. Therefore, S D(Ĥ). Moreover, since f(ˆx)2 rˆxˆxˆxˆx s, f(ˆx)ϕ 2 r ˆxˆxϕ 2 s ϕ 2. (54) As such, from Eq. (47), Eq. (53) is saisfied. If ϕ S, hen (f(ˆx)ϕ) L 2. Since, ±i[ĥ, Ô] cô (55)

8 8 as quadraic forms on C, we hus have for consan c. ±i[ĥ, Ô] = ±i{[ˆpˆp, ˆxˆx] [f(ˆx), ˆpˆp]} = ±{2(ˆp ˆx ˆx ˆp) (ˆp f(ˆx) f(ˆx) ˆp)} 2(ˆpˆp ˆxˆx) ˆpˆp ( f(ˆx)) 2 2(ˆpˆp ˆxˆx) ˆpˆp 2(a 2ˆxˆx b 2 ) cô, (56) Theorem 3. The eigenvalues of he Hamilonian D. PT -symmeric Bender-Brody-Müller Hamilonian iĥ = are imaginary, where ˆp = i x, =, and ˆx = x. i e iˆp (ˆxˆp ˆpˆx)( e iˆp ) (57) Corollary 3.. [] Soluions o he equaion iĥψ = Eψ are given by he Hurwiz zea funcion on he posiive half line R wih eigenvalues i(2z ). Proof. Le ψ be an eigenfuncion of Eq. (57) wih an eigenvalue λ = i(2z ): ψ z (x) = ζ(z, x ) = (x n) z (58) iĥψ = λψ. (59) Then we have he relaion Leing i e iˆp (ˆxˆp ˆpˆx)( e iˆp )ψ = λψ. (6) and insering Eq. (6) ino Eq. (6) wih ˆp = i x, =, and ˆx = x, we obain ϕ = [ exp( x )]ψ, (6) [x x x x]ϕ = λϕ. (62) Then we have (x x ϕ) ϕdx ( x xϕ) ϕdx = λ ϕ ϕdx. (63) Now we inegrae he firs erm on he LHS of Eq. (63) by pars o obain xϕ x ϕ dx = ϕ ϕdx ϕ x d (ϕ)dx, (64) dx and he second erm on he LHS of Eq. (63) by pars o obain xϕ x ϕdx = ϕϕ dx ϕx d dx (ϕ )dx. (65)

9 Upon subsiuing Eqs. (64) and (65) ino Eq. (63), we obain ϕ x d dx (ϕ)dx ϕx d dx (ϕ )dx = (λ 2)N, (66) 9 where N = ϕ ϕdx. (67) Nex, we spli ϕ ino real and imaginary componens, such ha ϕ = ϕ R iϕ I, (68) and subsiue Eq. (68) ino Eq. (66) such ha ϕ R x d dx ϕ Rdx ϕ I x d dx ϕ Idx N = λ 2. (69) Upon seing λ = i(2z ), Eq. (69) can be wrien ϕ R x d dx ϕ Rdx ϕ I x d dx ϕ Idx N = i( 2z). (7) 2 I can be seen ha all erms on he LHS of Eq. (69) are real, hereby verifying Theorem 3. Q.E.D. III. CONCLUSION In his noe, we have discussed he domain and eigenvalues of he BBM Hamilonian. [] Bender, C.M., Brody, D.C. and Mller, M.P., 26. Hamilonian for he zeros of he Riemann zea funcion. arxiv preprin arxiv: [2] Riemann, B., On he Number of Prime Numbers less han a Given Quaniy.(Ueber die Anzahl der Primzahlen uner einer gegebenen Grösse.). [3] Berry, M.V. and Keaing, J.P., 999. H= xp and he Riemann zeros. In Supersymmery and Trace Formulae (pp ). Springer US. [4] Connes, A., 999. Trace formula in noncommuaive geomery and he zeros of he Riemann zea funcion. Seleca Mahemaica, New Series, 5(), pp [5] Odlyzko, A.M., 2. The -nd zero of he Riemann zea funcion. Dynamical, Specral, and Arihmeic Zea Funcions: AMS Special Session on Dynamical, Specral, and Arihmeic Zea Funcions, January 5-6, 999, San Anonio, Texas, 29, p.39. [6] Faris, W.G. and Lavine, R.B., 974. Commuaors and self-adjoinness of Hamilonian operaors. Communicaions in Mahemaical Physics, 35(), pp [7] Glimm, J. and Jaffe, A., 972. The λϕ 24 Quanum Field Theory wihou Cuoffs. IV. Perurbaions of he Hamilonian. Journal of Mahemaical Physics, 3(), pp