Sep 14, 2018 the riemann hypothesis states that all zeros of the riemann zeta function are either negative even integers or complex numbers with real part equal to 12. Zeros of riemann zeta function university of chicago. Hamiltonian of a physical system and, therefore, the key to the proof of the riemann hypothesis may have been coded in physics 3. Wu and sprung 6, generated a onedimensional, therefore integrable, quantum mechanical model which can possess the riemann zeta zeros as energy eigenvalues and show the same level repulsion as that observed in. In the theory of the riemann zeta function, the set s. Comment on hamiltonian for the zeros of the riemann zeta. A proposed hamiltonian operator for riemann hypothesis. The main motivation for this is a similarity between the formulas that count the eigenvalues of a hamiltonian and those that count the zeros of the riemann zeta function. Brody department of mathematics, brunel university london, uxbridge ub8 3ph, united kingdom dorje. In 1859, bernhard riemann published a paper showing how to analytically continue this function to the entire complex plane, giving a holomorphic function everywhere except for a simple pole at s1. Hamiltonian for the zeros of the riemann zeta function surrey. Hamiltonian for the zeros of the riemann zeta function brunel. In mathematics, the riemann hypothesis is a conjecture that the riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 2. The search for a hamiltonian whose energy spectrum.
Louis, missouri 63, usa 2department of mathematics, brunel university london, uxbridge ub8 3ph, united kingdom. The details of this really require complex analysis. Its possible that there is a quantum chaotic hamiltonian. Hamiltonian for the riemann zeros consider the hamiltonian operator h. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number. The classical limit of hover is 2xp, which is consistent with the berrykeating conjecture. The boundary wavefunctions, giving rise to the riemann zeros, are found using the riemann siegel formula of the zeta function. A hamiltonian operator h is constructed with the property that if the eigenfunctions obey a suitable boundary. Sierra, the riemann zeros as energy levels of a dirac fermion in a potential built from the prime numbers in rindler spacetime, j. Hamiltonian h in 1 formally satis es the conditions of the hilbertp olya conjecture. Apr 09, 2017 this comment about the article hamiltonian for the zeros of the riemann zeta function, by c. Namely, we construct a supersymmetric quantum mechanical model whose energy eigenvalues correspond to the riemann zeta function in the strip 0 zeros on the critical line arise naturally from the vanishing ground state. The idea is then to transfer the question about the zeros of the zeta function to questions about the spectrum of an operator. On the zeros of the riemann zeta function in the critical strip by richard p.
As a toy model of this method, let be the sturmliouville problem. A note on hamiltonian for the zeros of the riemann zeta. Riemann zeta function nontrivial zeros because of the di. Zeros of the derivatives of the riemann zeta function ki, haseo and lee, yoonbok, functiones et approximatio commentarii mathematici, 2012. On repeated values of the riemann zeta function on the critical line banks, william d. Nonhermitian hamiltonian and the zeros of the riemann zeta.
This is as far as one gets when considering chaotic quantum oscillator given by hamiltonian h px so far. Nonhermitian hamiltonian and the zeros of the riemann zeta function dorje c. We present a spectral realization of the riemann zeros based on the propagation of a massless dirac fermion in a region of rindler spacetime and under the action of delta function potentials localized on the square free integers. Many consider it to be the most important unsolved problem in pure mathematics bombieri 2000. The riemann zetafunction s is a meromorphic function. The main contribution of this paper is to achieve the proof of riemann hypothesis. Comment on comment on hamiltonian for the zeros of the riemann zeta function authors. This comment about the article hamiltonian for the zeros of the riemann zeta function, by c. Introduction there is a conjecture, quoted to be made by hilbert and polya, that the zeros of the riemann zeta function. The riemann zeros as spectrum and the riemann hypothesis. Altmetric hamiltonian for the zeros of the riemann zeta. Muller4,5 1department of physics, washington university, st. Twin primes and the zeros of the riemann zeta function 3 twin ranks play the role of the primes p. Since it is a pole of first order, its principal value exists and is equal to the eulermascheroni constant.
Comment on comment on hamiltonian for the zeros of the. References 2 are devoted to the mathematical properties of the zeta function. David lyon the physics of the riemann zeta function. Pdf comment on comment on hamiltonian for the zeros of the. For the riemann zeta function on the critical line, see zfunction.
Pdf polymeric quantum mechanics and the zeros of the. Pdf a hamiltonian operator h is constructed with the property that if the eigenfunctions are formally required to obey a suitable boundary. The riemann zeta function and vacuum spectrum sergio jof. H is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues. Some local convexity theorems for the zeta function like. Find link is a tool written by edward betts longer titles found. The riemann zeta function or euler riemann zeta function. The analysis of 1 requires borel summation to regulate a divergent sum.
We will also see that an similar property of ls for. While h is not hermitian in the conventional sense, i h is p t symmetric with a. Riemanns outstanding hypothesis 1 that the nontrivial complex zeroes of the zetafunction. Quantization of nonstandard hamiltonians and the riemann zeros. The zeta function has a relation with the number of prime numbers less than a given quantity and the zeroes of zeta are deeply connected with the distribution of primes 1. Polymeric quantum mechanics and the zeros of the riemann zeta. Mueller submitted on 16 apr 2017 v1, last revised 27 sep 2017 this version, v2. Supersymmetry and the riemann zeros on the critical line. The zeros at the negative even integers are well understood and not mysterious, so people f. A spectral realization of the riemann zeros is achieved exploiting the symmetry of the model under the exchange of position and momenta which is related to the duality symmetry of the zeta function.
Bernhard riemann calculated the first six nontrivial zeros of the function and observed that they were all on the same straight line. That is, if the eigenfunctions of h are required to satisfy the boundary condition n0 0 for all n, then the eigenvalues fe nghave the property that f1 2 1 ie ngare the nontrivial zeros of the riemann zeta function. The riemann hypothesis is one of the most important open problems in mathematics, hilbert and polya 4 gave the conjecture that would exists an operator 1 2 il with l l so the eigenvalues of this operator would yield to the nontrivial zeros for the riemann zeta function, for the physicists one of the best candidates would be a. Pdf hamiltonian for the zeros of the riemann zeta function. The fact that riemann zeta function doesnt have a zero on res 1 is the most crucial step in the proof of the prime number theorem. Zeta functions for the riemann zeros centre mersenne. The classical limit of h is 2 x p, which is consistent with the berrykeating conjecture. The prime p j is the twin sieve analog of v x there. Calculating the nontrivial zeroes of the riemann zeta function is a whole entire field of mathematics. For t 1, the number of riemann zeros with imaginary part on the interval 0, t is given by 3 7 1 1 1 ln arg 2 2 8 2 t t n t o it e t. The second equality is a consequence of the unique factorization of integers into prime numbers. On the zeros of the riemann zeta function in the critical strip. Twin primes and the zeros of the riemann zeta function.
A hamiltonian operator h is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the riemann zeta function. While it was not created by riemann, it is named after him. Zeta function nontrivial zeros have real part equal to 12. The zeros of the zeta function could then be calculated the same way physicists calculate the possible energy levels for an electron in an atom, for example. Yeah, im jealous the riemann hypothesis is named after the fact that it is a hypothesis, which, as we all know, is the largest of the three sides of a right triangle. Quantum chaos, random matrix theory, and the riemann function. A general formula for the zeros of the analytic continuation formula of the riemann zeta function is obtained and it is shown that these zeros will always be real.
Sep 21, 2017 i gave you a probable bold assumption, let proton oscillate between up and down quark231 all of possible combination of n proton is their energy levelas merten conjecture, match nontrivial zero of zeta function by montgomery dyson con. Importance of the zeta function the zeta function is a very important function in mathematics. Hamiltonian for the zeros of the riemann zeta function. Riemann hypothesis is a conjecture that real part of every nontrivial zero of the riemann zeta function is 12. Its possible that there is a quantum chaotic hamiltonian which gives rise to the zeros of the zeta function. A hamiltonian operator hover is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the riemann zeta function. One of integral representations of zeta function valid for any s from the critical strip. What is the connection of the riemann zeta function to. Quantum mechanics and riemann hypothesis cern indico. The corresponding hamiltonian admits a selfadjoint extension that is tuned to the phase of the zeta function, on the critical line, in order to obtain the riemann. From the zeros of the riemann zeta function to its analytical.
We propose a new way of studying the riemann zeros on the critical line using ideas from supersymmetry. Polymeric quantum mechanics and the zeros of the riemann. How does the riemann hypothesis relate to quantum physics. This work proposes to investigate certain meromorphic functions defined by dirichlet series over the nontrivial zeros p of the riemann zeta function s, and to thoroughly compile their explicit analytical features. Nonhermitian hamiltonian and the zeros of the riemann. The riemann zeta function is eventually obtained through this. M o n t g o m e r y, zeros of the derivative of the riemann zeta function, acta math. The riemann hypothesis is an unproven statement referring to the zeros of the riemann zeta function. Introduction one of the most promising approaches to prove the riemann hypothesis 17 is based on the conjecture, due to polya and hilbert, that the riemann zeros are the eigenvalues of a quantum mechanical hamiltonian 8. Louis, missouri 63, usa 2department of mathematics, brunel university london, uxbridge ub8 3ph, united kingdom 3department of optical physics and modern natural science, st. It is of great interest in number theory because it implies results about the distribution of prime numbers. A schrodinger equation for solving the riemann hypothesis.
Hamiltonian for the zeros of the riemann zeta function carl m. A complete riemann zeta distribution and the riemann hypothesis nakamura, takashi, bernoulli, 2015. A note on hamiltonian for the zeros of the riemann zeta function authors. Further support came in the 1970s and 80s from the works of montgomery and odlyzko. The riemann zeta function plays a pivotal role in analytic number theory and has applications. Polymeric quantum mechanics and the zeros of the riemann zeta function 7 conditions are directly related to the energy spectrum through the selfadjoint extension of the hamiltonian operat or. The boundary wavefunctions, giving rise to the riemann zeros, are found using the riemannsiegel formula of the zeta function. In a report published in 1859, riemann stated that this might very well be.
The search for a hamiltonian whose energy spectrum coincides. Particular values of the riemann zeta function wikipedia. Many of the results about s that can be found with the usual tools of complex analysis have very important consequences on number theory see 4, p. The riemann hypothesis states that all zeros of the riemann zeta function are either negative even integers or complex numbers with real part equal to 12. The aim of these lectures is to provide an intorduction to the theory of the riemann zeta function for students who might later want to do research on the subject. Hamiltonian for the zeros of the riemann zeta function core. Quantum chaos, random matrix theory, and the riemann. Namely, we construct a supersymmetric quantum mechanical model whose energy eigenvalues correspond to the riemann zeta function in the strip 0 the complex parameter space and show that the zeros on the critical line arise naturally from the vanishing ground state energy. Particular values of the riemann zeta function, proof of the euler product formula for the riemann zeta function searching for riemann zeta function 43 found 359 total alternate case.
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