Geometric methods

We construct an explicit self-adjoint operator A∞ on a suitable Hilbert space \(H = L^2((0, ∞), dx), whose spectrum corresponds exactly to the imaginary parts γn of the nontrivial zeros of the Riemann zeta function. The operator takes the form A∞:= i d/dx + iβ/x, β = ½ with domain defined under appropriate boundary conditions. The eigenfunctions ψn(x) = Cn x-1/2 e-iγnx form a complete orthonormal system, assuming completeness. Under these assumptions, the Riemann Hypothesis follows.

This paper proposes a revolutionary unified game theory framework that integrates the three-solution decision system (Maximal Solution, Optimal Solution, Benevolent Solution), PanBoard cross-board universal algorithm, MWC→ES mapping theory, GoWulff geometric optimization model, and dynamic rule game theory. This framework breaks through traditional game theory's static assumptions, establishing a complete theoretical system from local winning configurations to global essential solutions, from static rules to dynamic evolution. Core contributions include: (1) Mathematical unified expression of the three-solution framework and dynamic switching mechanisms; (2) Cross-board strategy construction algorithms based on topological invariance; (3) Deep integration of geometric intuition with game logic; (4) Meta-game theory under rule evolution environments. Empirical analysis shows that this framework not only demonstrates performance surpassing traditional AI in closed games like Go, but also provides systematic decision guidance in open systems such as politics, business, and military. This research provides new theoretical foundations for artificial intelligence, decision science, complex system management, and other fields, marking a fundamental leap in game theory from "how to win games" to "how to dominate game rules.

The Prime Number Theorem tells us π(N) ∼ N / ln(N), but rarely provides systematic explanations for why prime density necessarily tends to zero. This paper proposes five mathematical principles of prime rarefaction: the Multiplicative Closure Principle, Sieving Exclusion Principle, Density Decay Principle, Logarithmic Growth Principle, and Structural Rarity Principle, establishing a complete logical chain from these first principles to asymptotic behavior. We not only prove the inevitability of prime density approaching zero but also reveal the deep mathematical mechanisms underlying this rarefaction. Through comparison with classical proofs, this paper provides a structural, first-principles-based complementary perspective, offering new insights into understanding the ultimate fate of primes.

Based on the theoretical framework of "Mathematical Relativity," this paper reveals a fundamental geometric characteristic of prime distribution: under logarithmic coordinate systems, the growth of cumulative prime averages exhibits a surprisingly perfect linear relationship. We prove that the geometric slope m of this line converges asymptotically to 1 as the observational scale increases. This discovery reveals that the average growth behavior of primes follows a fundamental power-law relationship, extending prime theory from the purely algebraic domain to the geometric domain and establishing the new research direction of "Prime Geometry." The complex distribution of primes, under the correct observational framework of "averaging" and "logarithmization," returns to the most concise geometric form. By treating the prime average curve as the hypotenuse of geometric triangles, we developed a trinity prediction algorithm combining geometric deduction, algebraic reduction, and structural calibration. This not only provides new methods for precise positioning of individual primes, but also reveals that simple and eternal geometric laws exist in the depths of the prime universe.

The Recursive Inflationary Fractal Time (RIFT) framework unifies number theory, fractal geometry, and quantum gravity through the handling of divergent infinite prime products and analytic continuation. Utilizing adelic integration, zeta function regularization, and recursive entropy dynamics, RIFT attains a theoretically stable cosmological constant Λ = 1 despite inherent divergences. Numerical validations up to 60 primes confirm convergence to Λ ≈ 1 ± 10-101 , with precision scaling logarithmically with prime count. Observable predictions include fractal-modulated gravitational waves and Mersenne prime-induced dark matter hierarchies.

his paper presents a reduction of Riemann's functional equation to a Riemann-Hilbert boundary value problem. The integral Hilbert transforms, which emerge in solving this problem, facilitate the computation of precise lower bounds for the zeta function. This approach highlights the effective use of boundary value problems and integral transforms in understanding the intricacies of the zeta function.

This paper studies the Schrödinger operator with Morse potential V k (u) = 1 4 e 2u + ke u on a right half-line [u 0 , ∞), and determines the Weyl asymptotics of eigenvalues for constant boundary conditions at the endpoint u 0. In consequence it obtains information on the location of zeros of the Whittaker function W κ,µ (x), for fixed real parameters κ, x with x > 0, viewed as an entire function of the complex variable µ. In this case all zeros lie on the imaginary axis, with the possible exception, if κ > 0, of a finite number of real zeros which lie in the interval −κ < µ < κ. We obtain an asymptotic formula for the number of zeros N (T) = {ρ | W κ,ρ (x) = 0, |Im(ρ)| < T } of the form N (T) = 2 π T log T + 2 π (2 log 2 − 1 − log x)T + O(1). Parallels are observed with zeros of the Riemann zeta function.

The Riemann hypothesis is an unproven statement referring to the zeros of the Riemann zeta function. Bernhard Riemann calculated the first six non-trivial zeros of the function and observed that they were all on the same straight line. In a report published in 1859, Riemann stated that this might very well be a general fact. The Riemann hypothesis claims that all non-trivial zeros of the zeta function are on the the line x = 1/2. The more than ten billion zeroes calculated to date, all of them lying on the critical line, coincide with Riemann's suspicion, but no one has yet been able to prove that the zeta function does not have non-trivial zeroes outside of this line.

The Riemann&#39;s hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn=1/2+iλn. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss–Jacobi theta series, allows us to provide the proper framework to construct the well-defined algorithm to compute the density of zeros in the critical line, which would complement the existing formulas in the literature for the density of zeros in the critical strip. Geometric probability theory furnishes the answer to the difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this geometric probabilistic framework to compute the probability if the RH is true, we apply it directly to the the hyperbolic sine function sinh (s) case which obeys a trivial analog of the RH (the HSRH). Its zeros are equally spaced in the imaginary axis sn=0+inπ. The geometric probability to find a zero (and an infinity of zeros) ...

The Riemann hypothesis, stating that all nontrivial zeros of the Riemann zeta function have real parts equal to 1/2, is one of the most important conjectures in mathematics. In this paper we prove the Riemann hypothesis by adding an extra unbounded term to the traditional definition, extending its validity to Rez>0. The Stolz-Cesàro theorem is then used to analyse zeta(z)/zeta(1-z) as a ratio of complex sequences. The results are analysed in both halves of the critical strip (0<Rez<1/2, 1/2<Rez<1), yielding a contradiction when it is assumed that zeta(z)=0 in either of these halves.

The possible connection of Riemann's Hypothesis on the non trivial zeroes of the zeta function [(z) with the theory of dynamical systems, both quantum and classical, is discussed. The conjecture of the existence of an underlying integrable structure is analysed, resorting on the one hand to the link between Riemann's zeta function and the Selberg trace formula, on the other to the relation between the zeroes of C(z) and the Gauss unitary ensemble of random matrices, to whichthrough basic results on the twisted de Rham cohomologya holonomic system of completely integrable differential equations can be associated. The complexity of Riemann's zeta function {(z) makes it a relevant object in the theory of chaotic dynamical systems, both quantum and classical. Also Riemann's hypothesis whereby the non-trivial zeroes of ((2) are expected to be distributed along the line Rez = 3 in the complex z-plane has had some impact in physical applications. Gutzwiller [I] has shown how in the solutions to Schrodinger's equation over a smooth orientable, but not compact, space of constant negative curvature with the topology of a punctured torus, particle eigenmomenta are related to the imaginary part y, of the zeroes en = $ + iy, of' ((2) (((en) = 0). Also, following a conjecture by Bilbert and P6lya, Berry [Z], [3] suggested the possible interpretation of the 7,'s as the eigenvalues of a hypothetical quantum hamiltonian operator, whose spectrum would turn out to be chaotic. The origin of chaoticity and complexity of the Riemann function is not difficult to understand, based on its very definition [4], [5]:

The number N (E) of complex zeros of the Riemann zeta function with positive imaginary part less than E is the sum of a 'smooth' functionN (E) and a 'fluctuation'. Berry and Keating have shown that the asymptotic expansion ofN (E) counts states of positive energy less than E in a 'regularized' semi-classical model with classical Hamiltonian H = xp. For a different regularization, Connes has shown that it counts states 'missing' from a continuum. Here we show how the 'absorption spectrum' model of Connes emerges as the lowest Landau level limit of a specific quantum mechanical model for a charged particle on a planar surface in an electric potential and uniform magnetic field. We suggest a role for the higher Landau levels in the fluctuation part of N (E).

The Riemann&#39;s hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn = 1/2 + iλn. By constructing a continuous family of scaling-like operators involving the Gauss–Jacobi theta series and by invoking a novel [Formula: see text]-invariant Quantum Mechanics, involving a judicious charge conjugation [Formula: see text] and time reversal [Formula: see text] operation, we show why the Riemann Hypothesis is true. An infinite family of theta series and their Mellin transform leads to the same conclusions.

The Riemann’s hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn =1/2 +iλn. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss-Jacobi theta series, allows to provide the proper framework to construct the well defined algorithm to compute the probability to find a zero (an infinity of zeros) in the critical line. Geometric probability theory furnishes the answer to the very difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this geometric probabilistic framework to compute the probability if the RH is true, we apply it directly to the the hyperbolic sine function sinh(s) case which obeys a trivial analog of the RH (the HSRH). Its zeros are equally spaced in the imaginary axis sn =0+inπ. The geometric probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetr...

A proof of the Riemann’s hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D (k,l) in one dimension, and their respective eigenfunctions ψs(t), parameterized by continuous real indexes k and l. Orthogonality of the eigenfunctions is connected to the zeros of the Riemann zeta-function. Due to the fundamental Gauss-Jacobi relation and the Riemann fundamental relation Z(s ′ ) = Z(1 − s ′), one can show that there is a direct concatenation among the following symmetries, t goes to 1/t, s goes to β − s (β a real), and s ′ goes to 1 − s ′ , which establishes a one-to-one correspondence between the label s of one orthogonal state to a unique vacuum state, and a zero s ′ of the ζ. It is shown that the RH is a direct consequence of these symmetries, by arguing in particular that an exclusion of a continuum of the zeros of the Riemann zeta function results in the discrete set of the zeros l...

This paper outlines further properties concerning the fractional derivative of the Riemann ζ function. The functional equation, computed by the introduction of the Grünwald-Letnikov fractional derivative, is rewritten in a simplified form that reduces the computational cost. Additionally, a quasisymmetric form of the aforementioned functional equation is derived (symmetric up to one complex multiplicative constant). The second part of the paper examines the link with the distribution of prime numbers. The Dirichlet η function suggests the introduction of a complex strip as a fractional counterpart of the critical strip. Analytic properties are shown, particularly that a Dirichlet series can be linked with this strip and expressed as a sum of the fractional derivatives of ζ. Finally, Theorem 4.3 links the fractional derivative of ζ with the distribution of prime numbers in the left half-plane.

In the quantum adelic field (string) theory models, vacuum energy -- cosmological constant vanish. The other (alternative ?) mechanism is given by supersymmetric theories. Some observations on prime numbers, zeta -- function and fine structure constant are also considered.

A great deal of research has been and still is being devoted to the zeros of the Riemann Zeta function (RZF) that are in the critical strip and known as the nontrivial zeros of RZF. The Riemann Hypothesis (RH) states that these zeros are all located on the critical line . Although a large number of nontrivial zeros have proved to be located on the critical line through numerical computation methods, starting with Riemann’s manual computation of the first few zero, no analytical proof or disproof of RH has been found since its conjecture by Riemann in 1859. In this paper, we implement a novel analytical approach to RH based on optimization. This analysis tool proved successful in deriving some important scientific theories and laws. Such a success prompted us to use this tool to analytically derive the location of RZF nontrivial zeros in order to either prove or disprove the Riemann Hypothesis. This was achieved by formulating and solving the appropriate location optimization problem.

We present, using spectral analysis, a possible way to prove the Riemann&#39;s hypothesis (RH) that the only zeroes of the Riemann zeta-function are of the form s=1/2+i\lambda_n. A supersymmetric quantum mechanical model is proposed as an alternative way to prove the Riemann&#39;s conjecture, inspired in the Hilbert-Polya proposal; it uses an inverse eigenvalue approach associated with a system of p-adic harmonic oscillators. An interpretation of the Riemann&#39;s fundamental relation Z(s) = Z(1-s) as a duality relation, from one fractal string L to another dual fractal string L&#39; is proposed.

During the last three decades, the problem of evaluation of the determinants of the Laplacians on Riemann manifolds has received considerable attention by many authors. The functional determinant for the n-dimensional sphere S n with the standard metric has been computed in several ways. Here we aim at computing the determinants of the Laplacians on S n (n = 8, 9) by mainly using ceratin known closed-form evaluations of series involving Zeta function.

During the last three decades, the problem of evaluation of the determinants of the Laplacians on Riemann manifolds has received considerable attention by many authors. The functional determinant for the n-dimensional sphere S n with the standard metric has been computed in several ways. Here we aim at computing the determinants of the Laplacians on S n (n = 8, 9) by mainly using ceratin known closed-form evaluations of series involving Zeta function.

In this paper, fractal calculus, which is called Fα-calculus, is reviewed. Fractal calculus is implemented on fractal interpolation functions and Weierstrass functions, which may be non-differentiable and non-integrable in the sense of ordinary calculus. Graphical representations of fractal calculus of fractal interpolation functions and Weierstrass functions are presented.

A proof of the Riemann&#x27;s hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D(k,l) in one dimension, and their respec-tive eigenfunctions ψs(t), parameterized by ...

Berry and Keating conjectured that the classical Hamiltonian H = xp is related to the Riemann zeros. A regularization of this model yields semiclassical energies that behave, in average, as the non trivial zeros of the Riemann zeta function. However, the classical trajectories are not closed, rendering the model incomplete. In this paper, we show that the Hamiltonian H = x(p + 2 p /p) contains closed periodic orbits, and that its spectrum coincides with the average Riemann zeros. This result is generalized to Dirichlet L-functions using different self-adjoint extensions of H. We discuss the relation of our work to Polya's fake zeta function and suggest an experimental realization in terms of the Landau model.

In this note we discuss explicitly the structure of some simple setsof values on the critical line (the rivial critical values&quot;) which are associatedwith the mean staircases emerging from the Zeta function. They are givenas solutions of an equation involving the Lambert W-function. The argumentof the latter function may then be set equal to a special N N (classical)matrix (for every N) related to the Hamiltonian of the Mehta-Dyson model.In this way we specify a function of an hermitean operator whose eigenvaluesare exactly these values. In the general case, the sets of such trivial criticalvalues (zeros) are special solutions of a parametric equation involving a linearcombination of Re (s) and Im (s) on the critical line.

We investigate the spectral zeta function of a self-similar Sturm-Liouville operator associated with a fractal self-similar measure on the half-line and C. Sabot's work connecting the spectrum of this operator with the iteration of a rational map of several complex variables. We obtain a factorization of the spectral zeta function expressed in terms of the zeta function associated with the dynamics of the corresponding renormalization map, viewed as a rational function on the complex projective plane, P 2 (C). The result generalizes to several complex variables and to the case of fractal Sturm-Liouville operators a factorization formula obtained by the second author for the spectral zeta function of a fractal string and later extended to the Sierpinski gasket and some other decimable fractals by A. Teplyaev. As a corollary, in the very special case when the underlying self-similar measure is Lebesgue measure on [0, 1], we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial in P 2 (C), thereby extending to several variables an analogous result by A. Teplyaev. The above fractal Hamiltonians and their spectra are relevant to the study of diffusions on fractals and to aspects of condensed matters physics, including to the key notion of density of states.

We study generalised prime systems P (1 < p 1 ≤ p 2 ≤ • • • , with p j ∈ R tending to infinity) and the associated Beurling zeta function ζ P (s) = ∞ j=1 (1 − p −s j) −1. Under appropriate assumptions, we establish various analytic properties of ζ P (s), including its analytic continuation and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of ζ P (s). Further we study 'well-behaved' g-prime systems, namely, systems for which both the prime and integer counting function are asymptotically well-behaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on N 2. Some of the above results may be relevant to the second author's theory of 'fractal membranes', whose spectral partition functions are precisely given by Beurling zeta functions. §1. Introduction 1.1 Generalised primes and Beurling zeta functions A generalised prime system P is a sequence of positive reals p 1 , p 2 , p 3 ,. .. satisfying 1 < p 1 ≤ p 2 ≤ • • • ≤ p n ≤ • • • and for which p n → ∞ as n → ∞. From these can be formed the system N of generalised integers or Beurling integers; that is, the numbers of the form p a 1 1 p a 2 2. .. p a k k where k ∈ N and a 1 ,. .. , a k ∈ N 0. 2 For simplicity, we shall often just refer to g-primes and g-integers. This system generalises the notion of prime numbers and the natural numbers obtained from them. Such systems were first introduced by Beurling [4] and have been

We propose a new way of studying the Riemann zeros on the critical line using ideas from supersymmetry. Namely, we construct a supersymmetric quantum mechanical model whose energy eigenvalues correspond to the Riemann zeta function in the strip 0 < Re s < 1 (in the complex parameter space) and show that the zeros on the critical line arise naturally from the vanishing ground state energy condition in this model.

In the quantum adelic field (string) theory models, vacuum energy -- cosmological constant vanish. The other (alternative ?) mechanism is given by supersymmetric theories. Some observations on prime numbers, zeta -- function and fine structure constant are also considered.

We comment on some apparently weak points in the novel strategies recently developed by various authors aiming at a proof of the Riemann hypothesis. After noting the existence of relevant previous papers where similar tools have been used, we refine some of these strategies. It is not clear at the moment if the problems we point out here can be resolved rigorously, and thus a proof of the RH be obtained, along the lines proposed. However, a specific suggestion of a procedure to overcome the encountered difficulties is made, what constitutes a step towards this goal.

An ensemble of 2×2 pseudo-Hermitian random matrices is constructed that possesses real eigenvalues with level-spacing distribution exactly as for the Gaussian unitary ensemble found by Wigner. By a re-interpretation of Connes&#39; spectral interpretation of the zeros of Riemann zeta function, we propose to enlarge the scope of search of the Hamiltonian connected with the celebrated Riemann hypothesis by suggesting that the Hamiltonian could also be PT-symmetric (or pseudo-Hermitian).

The distribution of prime numbers is directly related to the statistical distribution of the nontrivial zeros of the Riemann Zeta function that closely resembles that of energy levels of atomic nuclei. Moreover, Riemann Zeta function plays a fundamental role in many areas of mathematics, from number theory to geometry and theory of dynamical systems and in physics from quantum chaos to the theory of quantum fields and of quasicrystals. Unfortunately, no proof exists of the so-called Riemann hypothesis stating that all its nontrivial zeros lie on the critical line ℜ(s) = ½. A new method is proposed to prove the Riemann hypothesis based on the Hilbert-Pólya conjecture and a superconducting-type Hamiltonian in the Hilbert-Fock L2 space.

The Riemann&#39;s hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn=1/2+iλn. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss–Jacobi theta series, allows us to provide the proper framework to construct the well-defined algorithm to compute the density of zeros in the critical line, which would complement the existing formulas in the literature for the density of zeros in the critical strip. Geometric probability theory furnishes the answer to the difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this geometric probabilistic framework to compute the probability if the RH is true, we apply it directly to the the hyperbolic sine function sinh (s) case which obeys a trivial analog of the RH (the HSRH). Its zeros are equally spaced in the imaginary axis sn=0+inπ. The geometric probability to find a zero (and an infinity of zeros) ...

We comment on some apparently weak points in the novel strategies recently developed by various authors aiming at a proof of the Riemann hypothesis. After noting the existence of relevant previous papers where similar tools have been used, we refine some of these strategies. It is not clear at the moment if the problems we point out here can be resolved rigorously, and thus a proof of the RH be obtained, along the lines proposed. However, a specific suggestion of a procedure to overcome the encountered difficulties is made, what constitutes a step towards this goal.

The distribution of prime numbers is directly related to the statistical distribution of the nontrivial zeros of the Riemann Zeta function that closely resembles that of energy levels of atomic nuclei. Moreover, Riemann Zeta function plays a fundamental role in many areas of mathematics, from number theory to geometry and theory of dynamical systems and in physics from quantum chaos to the theory of quantum fields and of quasicrystals. Unfortunately, no proof exists of the so-called Riemann hypothesis stating that all its nontrivial zeros lie on the critical line ℜ(s) = ½. A new method is proposed to prove the Riemann hypothesis based on the Hilbert-Pólya conjecture and a superconducting-type Hamiltonian in the Hilbert-Fock L2 space.

 ABSTRACT: We review the Wu-Sprung potential adding a correction involving a fractional derivative of Riemann Zeta function, we study a global semiclassical analysis in order to fit a Hamiltonian H=T+V fitting to the Riemann zeros and another new Hamiltonian whose energy levels are precisely the prime numbers, through these paper we use the notation log ( ) ln( ) log( ) e x x x   for the logarithm , also unles we specify ( ) h    means that we sum over ALL the imaginary parts of the nontrivial zero on both the upper and lower complex plane.

The Riemann hypothesis states that the complex zeros of ζ(s) lie on the critical line Re(s) = 1 2 that is the non-imaginary solutions E n of ζ 1 2 + iE n = 0 are all real. George Pólya suggested a physical approach to prove the Riemann hypothesis. Also David Hilbert is associated with this conjecture, although there is no evidence he conjectured it. The conjecture simply states that if ρ n = 1 2 + it n are the non-trivial zeroes of the Riemann zeta function, then the t n 's correspond to the eigenvalues of a hermitian operator. Properties of the Riemann operator that are suggested by the quantum analogy. Lets call this operator H. 1. H has a classical counterpart (the Riemann dynamics) corresponding to the Hamiltonian flow on a phase space. 2. The Riemann dynamics is chaotic that is unstable and bounded. 3. The Riemann dynamics does not have a time-reversal symmetry. 4. The Riemann dynamics is homogeneously unstable. 5. The classical periodic orbits of the Riemann dynamics have periods that are independent of energy E and given by multiples of logarithms of prime numbers that is T m.q = m log q (m = 1, 2, ...; q prime) and associated actions are S m,q = Em log q. 6. The Maslov phases associated with the orbits are also peculiar: they are all π. 7. The Riemann dynamics possesses complex periodic orbits (instantons) whose periods are T complex,m = imπ.

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn = 1/2 + iλn. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss-Jacobi theta series, allows to provide the proper framework to construct the well defined algorithm to compute the probability to find a zero (an infinity of zeros) in the critical line. Geometric probability theory furnishes the answer to the very difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this geometric probabilistic framework to compute the probability if the RH is true, we apply it directly to the the hyperbolic sine function sinh(s) case which obeys a trivial analog of the RH (the HSRH). Its zeros are equally spaced in the imaginary axis sn = 0 + inπ. The geometric probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model implementing the Hilbert-Polya proposal to prove the RH based by postulating a Hermitian operator that reproduces all the λn for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the fractal analog of the Comtet-Bandrauk-Campbell (CBC) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter β is one-half the fractal dimension (D = 1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form λn = nπ and which coincide with the imaginary parts of the zeros of the function sinh(s). Finally, we discuss the relationship to the theory of 1/f noise.

A short review of Schrödinger hamiltonians for which the spectral problem has been related in the literature to the distribution of the prime numbers is presented here. We notice a possible connection between prime numbers and centrifugal inversions in black holes and suggest that this remarkable link could be directly studied within trapped Bose-Einstein condensates. In addition, when referring to the factorizing operators of Pitkanen and Castro and collaborators, we perform a mathematical extension allowing a more standard supersymmetric approach.

"Presentamos una caracterización de la cota inferior d para la distanciamínima de códigos algebraico-geométricos unipuntuales sobre curvas castillo. Calculamos explícitamente esta cota en el caso de un semigrupo de Weierstrass generado por dos elementos consecutivos. En particular, obtenemos una caracterización más simple del valor exacto de la distancia mínima paracódigos Hermitianos"

Abstract: In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics–random matrix theory–provides the connection between the two fields.

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn = 1/2 + iλn. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss-Jacobi theta series, allows to provide the proper framework to construct the well defined algorithm to compute the probability to find a zero (an infinity of zeros) in the critical line. Geometric probability theory furnishes the answer to the difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this geometric probabilistic framework to compute the probability if the RH is true, we apply it directly to the the hyperbolic sine function sinh(s) case which obeys a trivial analog of the RH (the HSRH). Its zeros are equally spaced in the imaginary axis sn = 0 + inπ. The geometric probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model implementing the Hilbert-Polya proposal to prove the RH by postulating a Hermitian operator that reproduces all the λn for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the fractal analog of the Comtet-Bandrauk-Campbell (CBC) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter β is one-half the fractal dimension (D = 1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form λn = nπ and which coincide with the imaginary parts of the zeros of the function sinh(s). Finally, we discuss the relationship to the theory of 1/f noise.