PROOF OF RIEMANN S HYPOTHESIS. MSC 2010 : 11A41; 11M06; 11M26 Keywords: Euler; Riemann; Hilbert; Polya; conjecture

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1 PROOF OF RIEMANN S HYPOTHESIS ROBERT DELOIN Abstract. Riemann s hypothesis (859) is the conjecture stating that: The real part of every non trivial zero of Riemann s zeta function is /. The main contribution of this paper is to achieve the proof of Riemann s hypothesis. The key idea is to provide an Hamiltonian operator whose real eigenvalues correspond to the imaginary part of the non trivial zeros of Riemann s zeta function and whose existence, according to Hilbert and Pólya, proves Riemann s hypothesis. MSC 00 : A4; M06; M6 Keywords: Euler; Riemann; Hilbert; Polya; conjecture Contents. Introduction. Preliminary notes.. The Hilbert-Pólya statement 3. Proof of Riemann s Hypothesis 3.. Proof of Hilbert-Pólya statement 3.. Preparing the unconditional proof of Riemann s Hypothesis Unconditional proof of Riemann s hypothesis 5 References 7. Introduction In his book [] of 748, Leonhard Euler ( ) proved what is now named the Euler product formula. This product is the result of the infinite sum: n= n = s p P ( /ps ) for any integer variable s > where P is the infinite set of primes. In his article [] of 859, Riemann (86-866) extended the Euler definition to the complex variable s of the zeta function: ζ(s) = p P ( /ps ) for any complex variable s It is known that the trivial zeros of the function are the infinite set: {s } = { m} for all integers m > 0

2 ROBERT DELOIN Riemann s hypothesis can be seen as stating that: Probably, the infinite set of the non trivial zeros {s } of ζ(s) can be written: {s } = { + it n} where t n is real. This conjecture is the first point of the eighth unresolved problem (among 3) that Hilbert listed in 900 [3] as well as the second unresolved problem listed in 000 by The Clay Mathematics Institute [4].. Preliminary notes.. The Hilbert-Pólya statement. Circa 94, Hilbert et Pólya [5], independently from each other, have orally stated that Riemann s hypothesis would be proved if it could be shown that the imaginary parts t n of the non trivial zeros of the symmetrical xi function ξ(s) derived from ζ(s), corresponded to the real eigenvalues of an unbounded Hamiltonian operator (here named Ĥξ) for which we could write: () Ĥ ξ ψ k = E k ψ k which is an equation of quantum physics where E k stands for the kd-components in a kd-space of a physical energy, with k =. So, the first and unique purely mathematical clue that we have is that the operator Ĥξ should be a square matrix of infinite dimension with real eigenvalues. This means that it could be written: t n t Ĥ ξ = (t n ) = n all t t n n being real t n Proof of Riemann s Hypothesis As Hilbert-Pólya statement will be used to prove Riemann s Hypothesis, the complete proof will be established in two steps: The first one proves that Hilbert-Pólya statement is indeed a conditional proof of Riemann s Hypothesis. The second one establishes the unconditional proof of Riemann s Hypothesis. 3.. Proof of Hilbert-Pólya statement. Proof. By definition, a complex number s is written: s = x + iy where x and y are real and i = By changing the conventional system of coordinates (x, y) of the complex plane into the new one (x = x, y = y), these complex numbers can be written:

3 PROOF OF RIEMANN S HYPOTHESIS 3 s = x + iy in the new system or: s = ( x) + iy when using the change of coordinates. Condition. We suppose that the Ĥξ operator exists and that it contains the infinitely many real eigenvalues t n coming from the non trivial zeros s of ζ(s). Hypothesis. We then suppose that these non trivial zeros lie anywhere in the complex plane with the two exceptions that they cannot lie on the real axis x or x (reserved for trivial zeros s ), which gives: y 0 and y 0 nor on the conventional critical line x = axis y, which gives: that becomes the new imaginary x and x 0 Then, each non trivial zero s of ζ(s) could be written: s = x + iy with x 0 and y 0 or, using the change of coordinates: s = ( x ) + iy with x and y 0 But using the fact that x = i x, they can be written: s = ( + i x ) + iy = + i(y + ix ) with x and y 0 or: s = + it with t = y + ix, x and y 0 and we get the result, as t = y + ix has to be real, that x has to be zero. This is not a direct contradiction to our hypothesis but this result has been proven wrong 0 3 times with the first 0 3 non trivial zeros s [6] for which x = /. Each of these 0 3 contradictions proves that our hypothesis is wrong and that Riemann s hypothesis is true conditionally to the existence of the Ĥξ operator, which is exactly Hilbert-Pólya statement. 3.. Preparing the unconditional proof of Riemann s Hypothesis. As Riemann s Hypothesis is now proven conditionally to the existence of the Ĥξ operator, we have to prove that the Ĥξ operator does exist. To do this, but first noticing that this operator refers only to the second set {s } of the non trivial zeros of ζ(s), we will consider the new and larger operator Ĥζ built with the zeros of both sets {s } and {s } as eigenvalues, an operator that also contains the real values t n (but not as eigenvalues):

4 4 ROBERT DELOIN Ĥ ζ = it it it As this new operator contains the real values t n, it enables us, at any time but if it exists, to rebuild the operator Ĥξ of Hilbert and Pólya. To simplify the writing, we set: (0) for all necessary zero values on and outside the diagonal, = ( m) and: + it it it 3... = ( + it ) n so that Ĥζ can be written: ( ) ( ) ( ) ( m) (0) Ĥ ζ = ( ( m) (0) (0) + it = ( (0) (0) n) (0) + i ) (0) Ĥ ξ But the matrices ( m) and ( + it n) representing the sets of zeros {s } and {s } can symbolically be replaced by their parametric form: m, m > 0 being an integer parameter + it n, t n being a real parameter The sets {s } and {s } can then be considered as the two infinite sets of roots of the polynomial of complex variable s: P (s) = (s s )(s s ) = s (s + s )s + s s P (s, m, t n ) = s ( m + + it n)s m( + it n) P (s, m, t n ) = s + (m ( + it n))s m( + it n) which, using matrices, can be written either: () P (s, m, t n ) = ( s s ) (m ( + it n)) m( + it n) or: (3) P (s, m, t n ) = ( ) (m ( + it n)) 0 s s 0 0 m( + it n)

5 PROOF OF RIEMANN S HYPOTHESIS 5 By setting: Ê k = ( s s ) (m ( + it n)) m( + it n) = ( s (m ( + it n))s m( + it n) ) and: ψ Ek = and by multiplying the first two matrices, equation () gives: (4) P (s, m, t n ) = ( s (m ( + it n))s m( + it n) ) = Êkψ Ek where k is now reduced to k = 3 so that our initial problem is also reduced to our 3D space. As by multiplying the first two matrices of (3), we also have: (5) P (s, m, t n ) = ( (m ( + it n)) m( + it n) ) s s = Ĥkψ Hk when we set: (6) Ĥ k = ( (m ( + it n)) m( + it n) ) ψ Hk = s and: s = ˆR = ˆRψ Ek where ˆR is the 3-dimensional transformation matrix from the orthogonal system of coordinates ψ Hk used to describe Ĥk to the orthogonal system ψ Ek used to describe Êk, and we have: Ĥ k ψ Hk = Ĥk ˆRψ Ek = Êkψ Ek Then, setting Ĥ = Ĥk ˆR, we get: (7) Ĥψ Ek = Êkψ Ek which is almost identical to equation (). Here, if almost identical is used, it is because Êk is an operator that may not be real as requied by equation (). That is why in the next section we look for the conditions that could make it real Unconditional proof of Riemann s hypothesis. Proof. From equation (7) we have: Ê k = Ĥ = Ĥk ˆR and as from equation (6), Ĥ k can also be written:

6 6 ROBERT DELOIN Ĥ k = ( ) (m ( + it n)) 0 = ( ) Â 0 0 m( + it n) when we set: (8) Â = (m ( + it n)) m( + it n) we get from (3) and () that: ( ) (9) Â s s = P (s, m, t n ) = ( s s ) Â But, for any s = x + iy, we have: P (s) = (s s )(s s ) = s (s + s )s + s s P (s, m, t n ) = s + ( m ( + it n) ) s m ( + it n = (x + iy) + ( m ( + it n) ) (x + iy) m ( + it n = ( x y + (m )x + yt ( n) + i xtn + y(m )) m mit n = ( x y + (m )x + yt n m ) + i ( xt n + y(x + m ) mt ) n and P (s, m, t n ) will be real only when: xt n + y(x + m ) mt n = 0 and so, only for the infinitely many curves in the complex plane such that: x + m x + m y = t n x + m = t n (x + m) + (x ) which, for x =, are all at y = t n and for x = m, are all at y = 0. Then, for all the points of all these curves (hyperboles), we have that: ( (0) P (s, m, t n ) curves = x y + (m ) )x + yt n m = V (x, y) is a real value and therefore the real mono-term matrix (V (x, y)) always verifies: () (V (x, y)) = (V (x, y)) = (V (x, y)) T where (V (x, y)) is the conjugate matrix of (V (x, y)) and (V (x, y)) T is the conjugate transpose of (V (x, y)). So, from (9), (0) and (), we can write: P (s, m, t n ) curves = V (x, y) = ( ) Â s ) ) s = ( s s ) Â which proves that the operator Â, also providing the real t n s to Ĥξ, verifies the equation of the observables in quantum physics, which is generally written: T

7 PROOF OF RIEMANN S HYPOTHESIS 7 < ψ  ψ > = ( < ψ  ψ > ) T where  is the Hamiltonian operator associated to the physical quantity A = V (x, y), < x and x > are the bra and ket operators on x, and ψ and ψ are the states of the physical quantity A before and after the measuring of A. As we have: Ê k = Ĥ = Ĥk ˆR = ( )  ˆR and as  can be associated with a physical quantity A = V (x, y), Ê k = Ĥ can also be associated with a physical quantity E in equation (7), this last one becoming identical to (). As we can rebuild the Hamiltonian operator Ĥξ linked to ζ(s) via the function P (s, m, t n ) and the existing operator Ĥ or Êk, this Hamiltonian operator Ĥ ξ does exist and as we have proven earlier that Riemann s hypothesis is true conditionally to the existence of the Ĥξ operator, Riemann s hypothesis is therefore unconditionally proven. Acknowledgements. This work is dedicated to my family. As I am a hobbyist, I wish to express my gratitude towards the Editors of this journal as well as towards the team of Reviewers for having welcomed, reviewed and accepted my article. References [] Euler L., Introductio in analysin infinitorum, 748. [] Riemann B., On the Number of Prime Numbers less than a Given Quantity, read on internet on February 5th, 06 at: [3] Hilbert, D., Mathematische Probleme, Göttinger Nachrichten, pp , 900. [4] CLay Mathematics Institute, Millennium Problems, read on internet on February 5th, 06 at: [5] Hilbert-Pólya conjecture, retrieved on December 9th, 06 on Wikipedia web page: conjecture #Possible connection with quantum mechanics [6] Gourdon X., Demichel P., Computation of zeros of the Zeta function, 004, retrieved on December 9th, 06 at: Robert Deloin address: rdeloin@free.fr