Four: Riemann conjecture German mathematician bernhard's Riemann conjecture? Riemann proposed it in 1859. It is an important and famous unsolved problem in mathematics (the crown of conjecture). Over the years, it has attracted many outstanding mathematicians to rack their brains for it. In 190 1, Helge von Koch pointed out that Riemann conjecture is equivalent to the prime number theorem with strong conditions. Now it has been verified that the prime numbers of 1500000000 are all valid for this theorem. But whether all the solutions of this theorem are valid or not has not been proved. Riemann conjecture is regarded as an important problem in contemporary mathematics, mainly because many in-depth and important mathematical and physical results can be proved on the premise of its establishment. Most mathematicians also believe that Riemann conjecture is correct (John? Enser? Littlewood and Selberg raised questions. Selberg partially changed his skeptical position in his later years. In a paper by 1989, he speculated that Riemann conjecture should also hold true for a wider class of functions. ) The Clay Institute of Mathematics has set up a prize of 1000000 dollars to reward the first person who is correctly proved.

Historical research

Riemann mentioned this famous conjecture in his 1859 paper, but this is not the central purpose of the paper, and he did not try to prove it. Riemann knows? The nontrivial zeros of a function are symmetrically distributed on a straight line s =? +it, and he knows that all its very zeros must be located in the region 0? Re(s)？ At 1. 1896, Jacques? Adama and Charles Jean Delaval? E-Poussin independently proved that there is no zero on the line Re(s) = 1. Together with other characteristics of extraordinary zeros that Riemann has proved, it shows that all extraordinary zeros must be in region 0.

Five: Young-Mills Existence and Mass Gap Young-Mills gauge field theory and mass gap is a basic problem of gauge field theory in theoretical physics. It is necessary to strictly prove the existence of Yang-Mills field theory in mathematics (that is, it must meet the standards of constructive quantum field theory) and prove that they have a mass gap, that is, the lightest single particle state predicted by the model is a positive mass. In 2000, Clay Institute of Mathematics provided a prize of1000000 yuan for seven Millennium mathematical problems, one of which was entitled Young-Mills gauge field theory and quality gap.

Background Most of the nontrivial 4-D quantum field theories we know have effective field theories with cut-off scales. Because the beta function of most models is positive, it seems that most of these models have a Landau pole, because we don't know whether they have extraordinary ultraviolet fixed points. Therefore, if such a quantum field theory is defined on every scale, it can only be a simple free field theory. However, Yang-Mills theory with non-commutative structural groups (without quarks) is an exception. It has a property called asymptotic freedom, which means it has a simple ultraviolet fixed point. Therefore, we can expect it to become an extraordinary and constructive four-dimensional quantum field model. The color confinement of the non-right Yi Qunyang-Mills theory has been proved to conform to the rigor of theoretical physics, but not to the rigor of mathematical physics [Note 3]. Basically, in other words, after the QCD scale (or the confinement scale here, because there are no quarks), which colored charged particles are chromodynamic? Flow tube? Are connected, so there is a linear potential between particles (? String? Tension x length). So it is impossible for free particles like gluons to exist. Without these limiting effects, we should see zero-mass gluons; However, because they are confined, we only see gluon beams bound without color charge. All plastic waves are mass, so we expect mass gap. The results of lattice gauge field theory make many workers believe that this model really has confinement phenomenon (due to the decrease of vacuum expectation of Wilson circle? Area rule? (area law), but this result does not conform to the rigor of mathematics.

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