e-book Generalized Riemann integral

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The professor Francis Villatoro. I am really grateful to him. He tries to divulge Science in Spain with his excellent blog written in Spanish language. He is a very active person in the world of Spanish Science and its divulgation. In his blog, he also tries to explain to the general public the latest news on HEP and other topics related with other branches of Physics, Mathematics or general Science.

It is not an easy task! His comments and remarks were incredibly useful for me, specially during my first logs. I had only some fuzzy ideas about what to do, what to write and, of course, I had no idea if I could explain stuff in a simple way while keeping the physical intuition and the mathematical background I wanted to include. His early criticism was very helpful, so this post is a tribute for him as well. After all, he suggested me the topic of this post! I encourage you to read him and his blog as long as you know Spanish or you can use a good translator.

Finally, let me express and show my deepest gratitude to John and Francis. Two great and extraordinary people and professionals in their respective fields who inspired and yet they do me in spirit and insight in my early and difficult steps of writing this blog. I am just convinced that Science is made of little, ordinary and small contributions like mine, and not only the greatest contributions like those making John and Francis to the whole world.

I wish they continue making their contributions in the future for many, many years yet to come. Now, let me answer the question Francis asked me to explain here with further details.

In general, it is a function of complex variable defined by the next equation:. The Riemann zeta function for real and entire positive values is a very well known and admired series by the mathematicians. Zeta values at even positive numbers are related to the Bernoulli numbers, and it is still lacking an analytic expression for the zeta values at odd positive numbers.

The Riemann zeta function over the whole complex plane satisfy the following functional equation:. Riemann zeta values are an example of beautiful Mathematics. From , then we have:. The harmonic series is divergent. The famous Euler result. Trivial zeroes of zeta. The first 13 Bernoulli numbers are:. For instance, , , and. Indeed, arises in string theory trying to renormalize the vacuum energy of an infinite number of harmonic oscillators. The result in the bosonic string is.

We also have that. The Riemann zeta value at the infinity is equal to the unit. Particularly important of this derivative are:. The alternating zeta function, called Dirichlet eta function, provides interesting values as well.

The Fundamental Theorem of Geometric Calculus via a Generalized Riemann Integral (1998)

Dirichlet eta function is defined and related to the Riemann zeta function as follows:. Special values of Dirichlet eta function are given by:. Remark I : is important in the physics realm, since the spectrum of the hydrogen atom has the following aspect.

Remark II : The fact that is finite implies that the energy level separation of the hydrogen atom in the Bohr level tends to zero AND that the sum of ALL the possible energy levels in the hydrogen atom is finite since is finite. If , we do know that is the case of the Kepler problem. Moreover, it is easy to observe that corresponds to tha harmonic oscillator, i.

We also know that is the infinite potential well. So the question is, what about a spectrum and so on? It is amazing how Riemann zeta function gets involved with a common origin of such a different systems and spectra like the Kepler problem, the harmonic oscillator and the infinite potential well! In other words, the equation or feynmanity has only the next solutions:.

The Riemann zeta function can be sketched on the whole complex plane, in order to obtain a radiography about the RH and what it means. The mathematicians have studied the critical strip with ingenious tools an frameworks. The now terminated ZetaGrid project proved that there are billions of zeroes IN the critical line.

No counterexample has been found of a non-trivial zeta zero outside the critical line and there are some arguments that make it very unlikely. If you want to know how the Riemann zeroes sound, M. Watkins has done a nice audio file to see their music. The next equation holds integer numbers, and non-trivial Riemann zeroes in the complex upper half-plane with :. This explicit formula was proved by Hadamard. The explicit formula follows from both product representations of , the Euler product on one side and the Hadamard product on the other side.

The function , sometimes written as , is the logarithmic integral. The explicit formula comes in some cool variants too. For instance, we can write. Date: January 3, That statement was never published formally, but it was remembered after all, and it was transmitted from one generation to another. However, Selberg, in the early s, proved a duality between the length spectrum of a Riemann surface and the eigenvalues of its Laplacian. Dialogue circa Dyson : Yes?

Montgomery : It seems the two-point correlations go as…. A step further was given in the s, by the mathematician Hugh Montgomery. The Riemann zeros tend not to cluster too closely together, but to repel. During a visit to the Institute for Advanced Study IAS in , he showed this result to Freeman Dyson, one of the founders of the theory of random matrices. These distributions are of importance in physics and mathematics. It is simple. The eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics.

Gaussian integral table

Subsequent work has strongly borne out the connection between the distribution of the zeros of the Riemann zeta function and the eigenvalues of a random Hermitian matrix drawn from the theory of the so-called Gaussian unitary ensemble, and both are now believed to obey the same statistics. The pair-correlation function of the zeros is given by the function:. This fact has therefore strengthened the analogy with the Selberg trace formula to the point where it gives precise statements.

However, the mysterious operator believed to provide the Riemann zeta zeroes remain hidden yet. However, some trials to guess the Riemann operator has been given from a semiclassical physical environtment as well. If that Berry-Keating conjecture is true. The simplest Hermitian operator corresponding to is. At current time, it is still quite vague, as it is not clear on which space this operator should act in order to get the correct dynamics, nor how to regularize it in order to get the expected logarithmic corrections.

Townsed, have conjectured that since this operator is invariant under dilatations perhaps the boundary condition for integer may help to get the correct asymptotic results valid for big. That it, in the large we should obtain. Indeed, the Berry-Keating conjecture opened another striking attack to prove the RH. Quantum chaos is the subject devoted to the study of quantum systems corresponding to classically chaotic systems. The Berry-Keating conjecture shed light further into the Riemann dynamics, sketching some of the properties of the dynamical system behind the Riemann Hypothesis.

The quantum hamiltonian operator behind the Riemann zeroes, in addition to the classical counterpart, the classical hamiltonian , has a dynamics containing the scaling symmetry. As a consequence, the trajectories are the same at all energy scale. The classical system corresponding to the Riemann dynamics is chaotic and unstable. The dynamics lacks time-reversal symmetry. The dynamics is quasi one-dimensional. A full dictionary translating the whole correspondence between the chaotic system corresponding to the Riemann zeta function and its main features is presented in the next table:.

The Riemannium arxiv paper was published later here: Reg. I remember myself reading the manuscript and being totally surprised. I was shocked during several weeks. I decided that I would try to understand that stuff better and better, and, maybe, make some contribution to it. The Spectrum of Riemannium was an amazing name, an incredible concept. So, I have been studying related stuff during all these years. And I have my own suspitions about what the riemannium and the zeta function are, but this is not a good place to explain all of them!

The riemannium is the mysterious physical system behind the RH. Its spectrum, the spectrum of riemannium, are given by the RH and its generalizations. For instance, the -dimensional regularized determinant is defined through the Riemann zeta function as follows:. It is ubiquitous in that approach, but, as far as I know, nobody has asked why is that issue important, as I have suspected from long time ago. Riemann zeta function is also used in the theory of Quantum Statistics. Quantum Statistics are important in Cosmology and Condensed Matter, so it is really striking that Riemann zeta values are related to phenomena like Bose-Einstein condensation or the Cosmic Microwave Background and also the yet to be found Cosmic Neutrino Background!

Let me begin with the easiest quantum indeed classical statistics, the Maxwell-Boltzmann MB statistics. In 3 spatial dimensions 3d the MB distribution arises we will work with units in which :. Usually, there are 3 thermodynamical quantities that physicists wish to compute with statistical distributions: 1 the number density of particles , 2 the energy density and 3 the pressure.

In the case of a MB distribution, we have the following definitions:. We can introduce the dimensionless variables ,. In this way,. This integral can be calculated in closed form with the aid of modified Bessel functions of the 2th kind:.

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And thus, we have the next results setting for simplicity :. These results can be simplified in some limit cases. For instance, in the massless limit. Moreover, we also know that. In such a case, we obtain:. We note that in this massless limit. Remark I : In the massless limit, and whenever there is no degeneracy, holds. Remark II : If there is a quantum degeneracy in the energy levels, i. For massless photons with helicity, there is a degeneracy.

Remark IV : Let us define as the number of ways an integer number can be expressed as a sum of the sth powers of integers. For instance,. If with and , then and the partition function is. Remark V : There are some useful integrals in quantum statistics. Introducing a scaled temperature , we get. Again, we can study a particularly simple case: the massless limit with. In this case, we get:. The FD distribution in 3d can be studied in a similar way.

Following the same approach as the BE distribution, we deduce that:. Remark I : For photons with degeneracy we obtain. The Cosmic Microwave Background is the relic photon radiation of the Big Bang, and thus it has a temperature due to photons in the microwave band of the electromagnetic spectrum.

Its value is:. Indeed, it also implies that the relic photon density is about. From theoretical Cosmology, it is related to the photon CMB temperature in the following way:. This temperature implies a relic neutrino density per species, i. Remark III : In Cosmology, for fermions in 3d note that BE implies , and that we must drop the factors in the next numerical values we can compute.

Remark IV : An example of the computation of degeneracy factor is the quark-gluon plasma degeneracy. Firstly we compute the gluon and quark degeneracies.

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In general, for charged leptons and nucleons , for neutrinos per species, of course , and for gluons and photons. Remember that massive particles with spin j will have. Remark V : For the Planck distribution, we also get the known result for the thermal distribution of the blackbody radiation. Remark VI : Sometimes the following nomenclature is used. Let us define the following shift operator :. Moreover, there is certain isomorphism between the shift operator space and the space of functions through the map. We define the generalized logarithm as the image under the previous map of.

That is:. Furthermore, the next contraints are also given for every generalized logarithm:. Group entropies are defined through the use of generalized logarithms. Define some discrete probability distribution with normalization. Therefore, the group entropy is the following functional sum:. It is called group entropy due to the fact that is connected to some universal formal group. This formal group will determine some correlations for the class of physical systems under study and its invariant properties.

In fact, the Tsallis logarithm itself is related to the Riemann zeta function through a beautiful equation! Under the Tsallis group exponential, the isomorphism is defined to be , and thus we easily get:. The eigenenergies or spectrum are given by and they have energies proportional to. Mathematically speaking,. What is this scale of energy? We do not know! Multi-particle states are defined in terms of the numbers of primons in the single-particle states :.

This corresponds to the factorization of into primes:. The statistical mechanics partition function IS, for the bosonic primon gas, the Riemann zeta function! The divergence of the zeta function at the value corresponding to the harmonic sum is due to the divergence of the partition function at certain temperature, usually called Hagedorn temperature.

The Hagedorn temperature is defined by:. This temperature represents a limit beyond the system of bosonic primons can not be heated up. To understand why, we can calculate the energy. A similar treatment can be built up for fermions rather than bosons, but here the Pauli exclusion principle has to be taken into account, i. Therefore can be 0 or 1 for all single particle state. As a consequence, the many-body states are labeled not by the natural numbers, but by the square-free numbers. The calculation is a bit more complex, but the partition function for a non-interacting fermion primon gas reduces to the relatively simple form.

The canonical ensemble is of course not the only ensemble used in statistical physics. Julia extended the Riemann gas approach to the grand canonical ensemble by introducing a chemical potential Julia, B. This generalisation of the Riemann gas is called the Beurling gas , after the Swedish mathematician Beurling who had generalised the notion of prime numbers.

Examining a boson primon gas with fugacity , it shows that its partition function becomes. Remarkable interpretation: pick a system, formed by two sub-systems not interacting with each other, the overall partition function is simply the product of the individual partition functions of the subsystems. From the previous equation of the free fermionic riemann gas we get exactly this structure, and so there are two decoupled systems. Although the divergence of the partition function hints the breakdown of the canonical ensemble, Julia also claims that the continuation across or around this critical temperature can help understand certain phase transitions in string theory or in the study of quark confinement.

The Riemann gas, as a mathematically tractable model, has been followed with much attention because the asymptotic density of states grows exponentially, , just as in string theory.

Numerical Integration Trapezoidal Rule

Moreover, using arithmetic functions it is not extremely hard to define a transition between bosons and fermions by introducing an extra parameter, called kappa , which defines an imaginary particle, the non-interacting parafermions of order. This order parameter counts how many parafermions can occupy the same state, i. Bowick, , J. Indeed, Bakas et al. This operation preserves the multiplicative property of the classically defined partition functions, i.

It is even more intriguing how interaction can be incorporated into the mixing by modifying the Dirichlet convolution with a kernel function or twisting factor. Using the unitary convolution Bakas establishes a pedagogically illuminating case, the mixing of two identical boson Riemann gases. He shows that. This result has an amazing meaning. Two identical boson Riemann gases interacting with each other through the unitary twisting, are equivalent to mixing a fermion Riemann gas with a boson Riemann gas which do not interact with each other. In this context, the fact that for square-free numbers is the manifestation of the Pauli exclusion principle itself!

In any QFT with fermions, is a unitary, hermitian, involutive operator where is the fermion number operator and is equal to the sum of the lepton number plus the baryon number, i. The action of this operator is to multiply bosonic states by 1 and fermionic states by This is always a global internal symmetry of any QFT with fermions and corresponds to a rotation by an angle.

This splits the Hilbert space into two superselection sectors. Bosonic operators commute with whereas fermionic operators anticommute with it. This operator really is, therefore, more useful in supersymmetric field theories. Remark III : the energy of the ground state is taken to be zero and the energy spectrum of the excited state is , where , , runs over the prime numbers. Let N and E denote now the number of particles in the ground state and the total energy of the system, respectively.

The fundamental theorem of arithmetic allows only one excited state configuration for a given energy. It immediately means that this gas preserves its quantum nature at any temperature, since only one quantum state is permitted to be occupied. The number fluctuation of any state even the ground state is therefore zero. In contrast, the changes in the number of particles in the ground state predicted by the canonical ensemble is a smooth non-vanishing function of the temperature, while the grand-canonical ensemble still exhibits a divergence.

This discrepancy between the microcanonical combinatorial and the other two ensembles remains even in the thermodynamic limit. However, we, physicists, think otherwise, since the spectrum does not increase with N more rapidly than , therefore the existence of a quantum mechanical potential supporting this spectrum is possible e.

And of course the question is: what kind of system has such an spectrum? Some temptative ideas for the potential based on elementary Quantum Mechanics will be given in the next section. Instead of considering the free Riemann gas, we could ask to Quantum Mechanics if there is some potential providing the logarithmic spectrum of the previous section.

Indeed, there exists such a potential. Equivalently, we could also define the spectrum of interacting riemannium gas as. From the physical viewpoint, the positive constant means repulsive interaction force. Then, using the turning point condition in this equation, we finally obtain. In summary, the logarithmic potential provides a model for the interacting Riemann gas!

Massive elementary particles with mass m can be understood as composite particles made of confined particles moving with some energy inside a sphere of radius R. We note that we do not define futher properties of the constituent particles e. These arguments and those I will give below can be found here.

Let us make the hypothesis that there is some force needed to counteract the centrifugal force. In Section 3, some generalizations of Hermite-Hadamard type inequalities for generalized beta r , g -preinvex functions via fractional integrals are given. These general inequalities give us some new estimates for the left-hand side of Gauss-Jacobi type quadrature formula and Hermite-Hadamard type fractional integral inequalities. Definition 2. Remark 2. In Definition 2. We next give new definition, to be referred as generalized beta r , g -preinvex function.

Also, for and we get the notion of generalized s, m -preinvex function see 3. In this section, in order to prove our main results regarding some new integral inequalities involving generalized beta r , g -preinvex functions, we need the following new Lemma:. Lemma 2. Theorem 2. Since is a generalized beta r , g -preinvex function on K , combining with Lemma 2. Corollary 2. Under the same conditions as in Theorem 2. Since f l is a generalized beta r, g -preinvex function on K, combining with Lemma 2. Hermite-Hadamard type fractional integral inequalities for generalized beta r, g -preinvex functions.

In this section, we prove our main results regarding some generalizations of Hermite-Hadamard type inequalities for generalized beta r, g -preinvex functions via fractional integrals. Theorem 3. Corollary 3. Under the same conditions as in Theorem 3. Remark 3. For different choices of positive values etc. Kashuri and R. Akkurt and H. Yildirim, On some fractional integral inequalities of Hermite-Hadamard type for r-preinvex functions, Khayyam J.

Du, J. Liao and Y. Li, Properties and integral inequalities of Hadamard-Simpson type for the generalized s, m -preinvex functions, J. Nonlinear Sci. Dragomir, J. Persson, Some inequalities of Hadamard type, Soochow J. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math. Antczak, Mean value in invexity analysis, Nonlinear Anal. Yang, X. Yang and K. Teo, Generalized invexity and generalized invariant monotonicity, J.

Theory Appl. Pini, Invexity and generalized convexity, Optimization. Kavurmaci, M. Avci and M. Chu, G. Wang and X. Zhang, Schur convexity and Hadamard's inequality, Math. Zhang, Y. Chu and X. Chu, M. Khan, T. Khan and T. Adil Khan, Y.