The zeros at the negative even integers are well understood and not mysterious, so people f. 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. Altmetric hamiltonian for the zeros of the riemann zeta. As a toy model of this method, let be the sturmliouville problem. 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.
Some local convexity theorems for the zeta function like. While it was not created by riemann, it is named after him. Hamiltonian for the zeros of the riemann zeta function surrey. The boundary wavefunctions, giving rise to the riemann zeros, are found using the riemannsiegel formula of the zeta function. 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. Importance of the zeta function the zeta function is a very important function in mathematics. 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. The boundary wavefunctions, giving rise to the riemann zeros, are found using the riemann siegel formula of the 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. Nonhermitian hamiltonian and the zeros of the riemann. Apr 09, 2017 this comment about the article hamiltonian for the zeros of the riemann zeta function, by c. Quantum chaos, random matrix theory, and the riemann. Introduction there is a conjecture, quoted to be made by hilbert and polya, that the zeros of the riemann zeta function. David lyon the physics of the riemann zeta function. Quantization of nonstandard hamiltonians and the riemann zeros.
Polymeric quantum mechanics and the zeros of the riemann. A complete riemann zeta distribution and the riemann hypothesis nakamura, takashi, bernoulli, 2015. 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. Louis, missouri 63, usa 2department of mathematics, brunel university london, uxbridge ub8 3ph, united kingdom. While h is not hermitian in the conventional sense, i h is p t symmetric with a. The riemann zeros as spectrum and the riemann hypothesis. Pdf hamiltonian for the zeros of the riemann zeta function. Zeros of riemann zeta function university of chicago. Hamiltonian h in 1 formally satis es the conditions of the hilbertp olya conjecture. We propose a new way of studying the riemann zeros on the critical line using ideas from supersymmetry. 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.
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. What is the connection of the riemann zeta function to. 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. Twin primes and the zeros of the riemann zeta function.
The riemann hypothesis is an unproven statement referring to the zeros of the riemann zeta function. Supersymmetry and the riemann zeros on the critical line. 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. Pdf a hamiltonian operator h is constructed with the property that if the eigenfunctions are formally required to obey a suitable boundary. Its possible that there is a quantum chaotic hamiltonian which gives rise to the zeros of the zeta function. Quantum chaos, random matrix theory, and the riemann function. The riemann zeta function and vacuum spectrum sergio jof. Quantum mechanics and riemann hypothesis cern indico. Pdf comment on hamiltonian for the zeros of the riemann. 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.
The prime p j is the twin sieve analog of v x there. In a report published in 1859, riemann stated that this might very well be. The riemann zeta function plays a pivotal role in analytic number theory and has applications. How does the riemann hypothesis relate to quantum physics. 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. The second equality is a consequence of the unique factorization of integers into prime numbers.
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. 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. 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. Twin primes and the zeros of the riemann zeta function 3 twin ranks play the role of the primes p. Louis, missouri 63, usa 2department of mathematics, brunel university london, uxbridge ub8 3ph, united kingdom 3department of optical physics and modern natural science, st. 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. The classical limit of hover is 2xp, which is consistent with the berrykeating conjecture. 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.
A hamiltonian operator h is constructed with the property that if the eigenfunctions obey a suitable boundary. We will also see that an similar property of ls for. Hamiltonian for the riemann zeros consider the hamiltonian operator h. Zeta function nontrivial zeros have real part equal to 12. The analysis of 1 requires borel summation to regulate a divergent sum. Brody department of mathematics, brunel university london, uxbridge ub8 3ph, united kingdom dorje. Polymeric quantum mechanics and the zeros of the riemann zeta. Hamiltonian for the zeros of the riemann zeta function. 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. H is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues. Comment on hamiltonian for the zeros of the riemann zeta.
The riemann zetafunction s is a meromorphic function. In the theory of the riemann zeta function, the set s. Zeros of the derivatives of the riemann zeta function ki, haseo and lee, yoonbok, functiones et approximatio commentarii mathematici, 2012. References 2 are devoted to the mathematical properties of the zeta function. This comment about the article hamiltonian for the zeros of the riemann zeta function, by c. Riemann zeta function nontrivial zeros because of the di. The riemann zeta function is eventually obtained through this. A proposed hamiltonian operator for riemann hypothesis. 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. Further support came in the 1970s and 80s from the works of montgomery and odlyzko.
On repeated values of the riemann zeta function on the critical line banks, william d. 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. Many consider it to be the most important unsolved problem in pure mathematics bombieri 2000. The idea is then to transfer the question about the zeros of the zeta function to questions about the spectrum of an operator. Pdf polymeric quantum mechanics and the zeros of the. 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 riemann zeta function or euler riemann zeta function. A note on hamiltonian for the zeros of the riemann zeta function authors. Muller4,5 1department of physics, washington university, st. Find link is a tool written by edward betts longer titles found. Mueller submitted on 16 apr 2017 v1, last revised 27 sep 2017 this version, v2. Hamiltonian for the zeros of the riemann zeta function brunel. Nonhermitian hamiltonian and the zeros of the riemann zeta function dorje c. 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.
The search for a hamiltonian whose energy spectrum coincides. Hamiltonian for the zeros of the riemann zeta function carl m. Pdf comment on comment on hamiltonian for the zeros of the. Riemanns outstanding hypothesis 1 that the nontrivial complex zeroes of the zetafunction. Bernhard riemann calculated the first six nontrivial zeros of the function and observed that they were all on the same straight line.
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. A schrodinger equation for solving the riemann hypothesis. It is of great interest in number theory because it implies results about the distribution of prime numbers. 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.
Riemann hypothesis is a conjecture that real part of every nontrivial zero of the riemann zeta function is 12. On the zeros of the riemann zeta function in the critical strip. 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. A note on hamiltonian for the zeros of the riemann zeta. For the riemann zeta function on the critical line, see zfunction. Sierra, the riemann zeros as energy levels of a dirac fermion in a potential built from the prime numbers in rindler spacetime, j. M o n t g o m e r y, zeros of the derivative of the riemann zeta function, acta math. The search for a hamiltonian whose energy spectrum. Calculating the nontrivial zeroes of the riemann zeta function is a whole entire field of mathematics. 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. Comment on comment on hamiltonian for the zeros of the. Hamiltonian of a physical system and, therefore, the key to the proof of the riemann hypothesis may have been coded in physics 3. The details of this really require complex analysis.
Nonhermitian hamiltonian and the zeros of the riemann zeta. Zeta functions for the riemann zeros centre mersenne. From the zeros of the riemann zeta function to its analytical. Comment on comment on hamiltonian for the zeros of the riemann zeta function authors. One of integral representations of zeta function valid for any s from the critical strip.
931 1046 580 1425 1113 242 1469 1551 668 731 688 39 235 1391 196 1487 1468 169 29 372 553 1070 1364 609 149 29 435 151 1279 584 1200 618 1358 1056 1440 912 781 110 84 1224 1176 1167 208 484