Hamiltonian for the zeros of the riemann zeta function pdf

The riemann zetafunction s is a meromorphic function. Hamiltonian for the zeros of the riemann zeta function surrey. The riemann hypothesis is an unproven statement referring to the zeros of the riemann zeta function. Further support came in the 1970s and 80s from the works of montgomery and odlyzko. Brody department of mathematics, brunel university london, uxbridge ub8 3ph, united kingdom dorje. Some local convexity theorems for the zeta function like. Pdf a hamiltonian operator h is constructed with the property that if the eigenfunctions are formally required to obey a suitable boundary. The idea is then to transfer the question about the zeros of the zeta function to questions about the spectrum of an operator. The search for a hamiltonian whose energy spectrum. Its possible that there is a quantum chaotic hamiltonian which gives rise to the zeros of the zeta function. Its possible that there is a quantum chaotic hamiltonian. Particular values of the riemann zeta function wikipedia.

On repeated values of the riemann zeta function on the critical line banks, william d. Apr 09, 2017 this comment about the article hamiltonian for the zeros of the riemann zeta function, by c. The analysis of 1 requires borel summation to regulate a divergent sum. The main contribution of this paper is to achieve the proof of riemann hypothesis. Calculating the nontrivial zeroes of the riemann zeta function is a whole entire field of mathematics. 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 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. Riemanns outstanding hypothesis 1 that the nontrivial complex zeroes of the zetafunction. Since it is a pole of first order, its principal value exists and is equal to the eulermascheroni constant. 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. Hamiltonian h in 1 formally satis es the conditions of the hilbertp olya conjecture.

Bernhard riemann calculated the first six nontrivial zeros of the function and observed that they were all on the same straight line. Altmetric hamiltonian for the zeros of the riemann zeta. 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. Nonhermitian hamiltonian and the zeros of the riemann zeta. Sierra, the riemann zeros as energy levels of a dirac fermion in a potential built from the prime numbers in rindler spacetime, j. Louis, missouri 63, usa 2department of mathematics, brunel university london, uxbridge ub8 3ph, united kingdom 3department of optical physics and modern natural science, st. Nonhermitian hamiltonian and the zeros of the riemann. References 2 are devoted to the mathematical properties of the zeta function. A note on hamiltonian for the zeros of the riemann zeta function authors.

We propose a new way of studying the riemann zeros on the critical line using ideas from supersymmetry. 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. 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. The classical limit of hover is 2xp, which is consistent with the berrykeating conjecture. The second equality is a consequence of the unique factorization of integers into prime numbers. A complete riemann zeta distribution and the riemann hypothesis nakamura, takashi, bernoulli, 2015. While it was not created by riemann, it is named after him. Zeta function nontrivial zeros have real part equal to 12. Hamiltonian for the zeros of the riemann zeta function.

Hamiltonian for the zeros of the riemann zeta function carl m. 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. We will also see that an similar property of ls for. Comment on hamiltonian for the zeros of the riemann zeta. David lyon the physics of the riemann zeta function. 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. Polymeric quantum mechanics and the zeros of the riemann. Twin primes and the zeros of the riemann zeta function 3 twin ranks play the role of the primes p. 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. Importance of the zeta function the zeta function is a very important function in mathematics. 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.

A schrodinger equation for solving the riemann hypothesis. The riemann zeta function plays a pivotal role in analytic number theory and has applications. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number. The riemann zeta function is eventually obtained through this. 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 prime p j is the twin sieve analog of v x there. What is the connection of the riemann zeta function to.

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. 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. For the riemann zeta function on the critical line, see zfunction. The riemann zeta function or euler riemann zeta function. 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. In a report published in 1859, riemann stated that this might very well be. Supersymmetry and the riemann zeros on the critical line. 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. On the zeros of the riemann zeta function in the critical strip by richard p. The riemann zeros as spectrum and the riemann hypothesis. Muller4,5 1department of physics, washington university, st. Many consider it to be the most important unsolved problem in pure mathematics bombieri 2000. 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. One of integral representations of zeta function valid for any s from the critical strip.

H is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues. Hamiltonian for the riemann zeros consider the hamiltonian operator h. Pdf comment on hamiltonian for the zeros of the riemann. 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. While h is not hermitian in the conventional sense, i h is p t symmetric with a. Pdf polymeric quantum mechanics and the zeros of the. A hamiltonian operator h is constructed with the property that if the eigenfunctions obey a suitable boundary. A note on hamiltonian for the zeros of the riemann zeta. The riemann zeta function and vacuum spectrum sergio jof. 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 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.

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. Zeros of the derivatives of the riemann zeta function ki, haseo and lee, yoonbok, functiones et approximatio commentarii mathematici, 2012. On the zeros of the riemann zeta function in the critical strip. M o n t g o m e r y, zeros of the derivative of the riemann zeta function, acta math. The zeros at the negative even integers are well understood and not mysterious, so people f. 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. In the theory of the riemann zeta function, the set s. Nonhermitian hamiltonian and the zeros of the riemann zeta function dorje c. A proposed hamiltonian operator for riemann hypothesis. Introduction there is a conjecture, quoted to be made by hilbert and polya, that the zeros of the riemann zeta function. Hamiltonian of a physical system and, therefore, the key to the proof of the riemann hypothesis may have been coded in physics 3.

Comment on comment on hamiltonian for the zeros of the. The classical limit of h is 2 x p, which is consistent with the berrykeating conjecture. Comment on comment on hamiltonian for the zeros of the riemann zeta function authors. Pdf hamiltonian for the zeros of the riemann zeta function. Quantum chaos, random matrix theory, and the riemann function. Quantum mechanics and riemann hypothesis cern indico. Hamiltonian for the zeros of the riemann zeta function core.

This is as far as one gets when considering chaotic quantum oscillator given by hamiltonian h px so far. Mueller submitted on 16 apr 2017 v1, last revised 27 sep 2017 this version, v2. The details of this really require complex analysis. Riemann hypothesis is a conjecture that real part of every nontrivial zero of the riemann zeta function is 12. 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.

As a toy model of this method, let be the sturmliouville problem. Zeta functions for the riemann zeros centre mersenne. How does the riemann hypothesis relate to quantum physics. The search for a hamiltonian whose energy spectrum coincides. 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. 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. 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 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. Hamiltonian for the zeros of the riemann zeta function brunel. Louis, missouri 63, usa 2department of mathematics, brunel university london, uxbridge ub8 3ph, united kingdom.

The boundary wavefunctions, giving rise to the riemann zeros, are found using the riemannsiegel formula of the zeta function. Polymeric quantum mechanics and the zeros of the riemann zeta. The boundary wavefunctions, giving rise to the riemann zeros, are found using the riemann siegel formula of the zeta function. This comment about the article hamiltonian for the zeros of the riemann zeta function, by c. Quantization of nonstandard hamiltonians and the riemann zeros. 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. Quantum chaos, random matrix theory, and the riemann. From the zeros of the riemann zeta function to its analytical. Pdf comment on comment on hamiltonian for the zeros of the. 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.