The Prime Sequence: Demonstrably Highly Organized While Also Opaque and Incomputable-With Remarks on Riemann’s Hypothesis, Partition, Goldbach’s Conjecture, Euclid on Primes, Euclid’s Fifth Postulate, Wilson’s Theorem along with Lagrange’s Proof of It and Pascal’s Triangle, and Rational Human Intelligence

The main design of this paper is to determine once and for all the true nature and status of the sequence of the prime numbers, or primes—that is, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and so on. The main conclusion revolves entirely around two points. First, on the one hand, it is shown that the prime sequence exhibits an extremely high level of organization. But second, on the other hand, it is also shown that the clearly detectable organization of the primes is ultimately beyond human comprehension. This conclusion runs radically counter and opposite—in regard to both points—to what may well be the default view held widely, if not universally, in current theoretical mathematics about the prime sequence, namely the following. First, on the one hand, the prime sequence is deemed by all appearance to be entirely random, not organized at all. Second, on the other hand, all hope has not been abandoned that the sequence may perhaps at some point be grasped by human cognition, even if no progress at all has been made in this regard. Current mathematical research seems to be entirely predicated on keeping this hope alive. In the present paper, it is proposed that there is no reason to hope, as it were. According to this point of view, theoretical mathematics needs to take a drastic 180-degree turn. The manner of demonstration that will be used is direct and empirical. Two key observations are adduced showing, 1), how the prime sequence is highly organized and, 2), how this organization transcends human intelligence because it plays out in the dimension of infinity and in relation to π. The present paper is part of a larger project whose design it is to present a complete and final mathematical and physical theory of rational human intelligence. Nothing seems more self-evident than that rational human intelligence is subject to absolute limitations. The brain is a material and physically finite tool. Everyone will therefore readily agree that, as far as reasoning is concerned, there are things that the brain can do and things that it cannot do. The search is therefore for the line that separates the two, or the limits beyond which rational human intelligence cannot go. It is proposed that the structure of the prime sequence lies beyond those limits. The contemplation of the prime sequence teaches us something deeply fundamental about the human condition. It is part of the quest to Know Thyself.

Physical entropy, information entropy and their evolution equations

Inspired by the evolution equation of nonequilibrium statistical physics entropy and the concise statistical formula of the entropy production rate, we develop a theory of the dynamic information entropy and build a nonlinear evolution equation of the information entropy density changing in time and state variable space. Its mathematical form and physical meaning are similar to the evolution equation of the physical entropy: The time rate of change of information entropy density originates together from drift, diffusion and production. The concise statistical formula of information entropy production rate is similar to that of physical entropy also. Furthermore, we study the similarity and difference between physical entropy and information entropy and the possible unification of the two statistical entropies, and discuss the relationship among the principle of entropy increase, the principle of equilibrium maximum entropy and the principle of maximum information entropy as well as the connection between them and the entropy evolution equation.

A critical case of Rallis inner product formula

Letπbe a genuine cuspidal representation of the metaplectic group of rank n.We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2n+1.We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands L-function ofπtwisted by a character.The bulk of this paper focuses on proving a case of regularised Siegel-Weil formula,on which the Rallis inner product formula is based and whose proof is missing in the literature.

The Spectrum of a Rational Function

Let k be an infinite field,A be a finite set of k,and Q∈k[x](with x=(x_(1),...,X_(n))and n≥2)be a noncons taut polynomial.The main goal of this paper is to construct a polynomial P(x)∈k[x]with suitably large partial degrees in x_(1),...,x_(n-1)such that P and Q axe coprime,and P-aQ is reducible for all a in A.