Nontrivial Zeros Research Articles

Behavior of Non-trivial Zeros and Prime Numbers in Riemann Hypothesis

Behavior of Non-trivial Zeros and Prime Numbers in Riemann Hypothesis

Two string theory flavours of generalised Eisenstein series

Generalised Eisenstein series are non-holomorphic modular invariant functions of a complex variable, τ, subject to a particular inhomogeneous Laplace eigenvalue equation on the hyperbolic upper-half τ-plane. Two infinite classes of such functions arise quite naturally within different string theory contexts. A first class can be found by studying the coefficients of the effective action for the low-energy expansion of type IIB superstring theory, and relatedly in the analysis of certain integrated four-point functions of stress tensor multiplet operators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills theory. A second class of such objects is known to contain all two-loop modular graph functions, which are fundamental building blocks in the low-energy expansion of closed-string scattering amplitudes at genus one. In this work, we present a Poincaré series approach that unifies both classes of generalised Eisenstein series and manifests certain algebraic and differential relations amongst them. We then combine this technique with spectral methods for automorphic forms to find general and non-perturbative expansions at the cusp τ → i∞. Finally, we find intriguing connections between the asymptotic expansion of these modular functions as τ → 0 and the non-trivial zeros of the Riemann zeta function.

High-order SUSY-QM, the quantum XP model and zeroes of the Riemann Zeta function

Abstract Making use of the first- and second-order algorithms of supersymmetric quantum mechanics (SUSY-QM), we construct quantum mechanical Hamiltonians whose spectra are related to the zeroes of the Riemann Zeta function ζ(s). Inspired by the model of Das and Kalauni (DK) A Das and P Kalauni (2019 Physics Letters B 791, 265-269) which corresponds to this function in the strip 0 < Re [ s ] < 1 , and taking the factorization energy equal to zero, we use the wave function ∣x∣−S , S ∈ C , as a seed solution for our algorithms, obtaining XP-like operators. Thus, we construct SUSY-QM partner Hamiltonians whose zero energy mode locates exactly the nontrivial zeroes of ζ(s) along the critical line Re [ s ] = 1 / 2 in the complex plane. We further find that unlike the DK case, where the SUSY-QM partner potentials correspond to free particles, our partner potentials belong to the family of inverse squared distance potentials with complex couplings.

An investigation into the distinctive properties of LiAuS and NaAuS topological insulators

Topological insulators (TIs) have garnered significant interest in condensed matter physics and materials science due to their unique properties. Recently, Half-Heusler compounds have been predicted to exhibit topological insulator behavior based on electronic structure. We conducted a study on potential alternatives for optoelectronic, photorefractive, and photovoltaic devices. Using density-functional-theory (DFT) calculations, a systematic comparison of the structural, electronic, and optical properties of LiAuS and NaAuS is performed. The geometrically optimized structures of cubic LiAuS and NaAuS were compared with theoretical values. The electronic band structure revealed a non-trivial zero band gap at the G-point, and the influence of different orbitals on the conduction band (CB) and valence band (VB) was analyzed. The optical properties including absorption, reflectivity, index of refraction, extinction coefficient, conductivity, and dielectric function were examined over the energy spectrum ranging from 0 to 30 eV. LiAuS displays a notable absorption peak at 7.95 eV, whereas NaAuS exhibits a peak at 6.79 eV. The reflectivity peak, on the other hand, is higher at 19.79 eV and 8.29 eV for LiAuS, and at 6.74 eV for NaAuS. The reflective indices for LiAuS and NaAuS are determined to be 3.5 and 7.03, respectively. It is observed that LiAuS and NaAuS possess elevated optical conductivity at 6.26 eV and 6.14 eV, respectively. The conductivities of LiAuS and NaAuS in terms of their visual properties have been claimed to be 8.0(1/fs) and 6.84(1/fs), individually. In view of its high dielectric constant, LiAuS exhibits significant absorption peaks at 0.91 eV, 4.63 eV, and 13.63 eV, whereas NaAuS displays peaks at 0.07 eV, 4.45 eV, and 13.12 eV. Owing to their optical characteristics, LiAuS and NaAuS possess the potential for implementation in photorefractive and optoelectronic devices.

Searching for nontrivial zeros in finite field carrying geometric metric ratios

With much emphasis on how to solve one of the toughest mathematical problems in field of prime number theory. There is a need to find a good coherent solution that gives much understanding into how to better unravel and solve the Riemann Hypothesis in the best way.

A symmetry perspective of the Riemann zeros

We study the relationship between the zeros of the Riemann zeta function and physical systems exhibiting supersymmetry, PT symmetry and SU(2) group symmetry. Our findings demonstrate that unbroken supersymmetry is associated with the presence of non-trivial zeros of the zeta function. However, for other cases, supersymmetry is spontaneously broken and the ground-state energy of the system is not zero. Moreover, we have established the manifestation of PT symmetry invariance within our supersymmetric system. In addition, our findings provide insights into a SU(2) symmetry that arises within these systems, with the Hilbert space having two-level structure.

Amplitude-like functions from entire functions

Recently a function was constructed that satisfies all known properties of a tree-level scattering of four massless scalars via the exchange of an infinite tower of particles with masses given by the non-trivial zeroes of the Riemann zeta function. A key ingredient in the construction is an even entire function whose only zeroes coincide with the non-trivial zeroes of the Riemann zeta function. In this paper we show that exactly the same conclusions can be drawn for an infinite class of even entire functions with only zeroes on the real line. This shows that the previous result does not seem to be connected to specific properties of the Riemann zeta function, but it applies more generally. As an application, we show that exactly the same conclusions can be drawn for L-functions other than the Riemann zeta function.

HOW MULTIPLE BRANCHES OF MATHEMATICS UNITE THE MOBIUS STRIP WITH MULTIPLE PHENOMENA IN PHYSICS AND COSMOLOGY

Question – How does the Riemann Hypothesis Support Topological Propulsion and Faster-than-light Travel? Answer – 
 
 a) Using the axiom that there indeed are infinitely many nontrivial zeros on the critical line (calculations have confirmed the hypothesis to be true to over 13 trillion places), the critical line is identified as the y-axis of Wick rotation. This suggests the y-axis is literally infinite and that infinity equals zero. In this case, it is zero distance in time and space. Travelling zero distance is done instantly and is therefore faster-than-light travel. 
 b) Wick rotation is essential to this article’s description of a topological (mathematical) universe and the Riemann hypothesis’ identification with Wick means the hypothesis doesn’t just apply to the distribution of prime numbers but also applies to the fundamental structure of the mathematical universe’s space-time. 
 
 An absolutely essential ingredient of time travel by applied maths is being able to journey into the remote past and future. For this reason, a paragraph about the coexistence of past, present, and future (without such coexistence, time travel could never be a possibility) is included in a section titled “Riemann Hypothesis and Wick Rotation Support Time Travel into the Past and Future”.
 This article provides insights into travelling at significant fractions of – as well as faster than - light, the Higgs boson and field, electroweak interaction, dark matter, dark energy, other dimensions, space-time (eg the expanding-universe question and time travel), quantum mechanics, quantum computers, the Riemann hypothesis, and Unidentified Flying Objects. 
 The article is also a brief summary discussing how the Matrix, the Riemann hypothesis, quaternions, Wick rotation, and imaginary numbers unite topology’s Mobius strip with known and unknown dimensions as well as the Higgs boson, Higgs field, dark matter, dark energy, antigravitons, and the static universe. The article also explains Unidentified Flying Objects as the result of future human technology.

Lower bounds for negative moments of ζ′(ρ)$\zeta ^{\prime }(\rho )$

AbstractWe establish lower bounds for the discrete 2kth moment of the derivative of the Riemann zeta function at nontrivial zeros for all under the Riemann hypothesis and the assumption that all zeros of are simple.

A Characterization of Nontrivial Zeros of Riemann Zeta Function

A Characterization of Nontrivial Zeros of Riemann Zeta Function

SUM OF VALUES OF THE IDEAL CLASS ZETA-FUNCTION OVER NONTRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION

AbstractWe prove an upper bound for the sum of values of the ideal class zeta-function over nontrivial zeros of the Riemann zeta-function. The same result for the Dedekind zeta-function is also obtained. This may shed light on some unproved cases of the general Dedekind conjecture.

Supersymmetric quantum mechanics and the Riemann hypothesis

We construct a supersymmetric quantum mechanical model in which the energy eigenvalues of the Hamiltonians are the products of Riemann zeta functions. We show that the trivial and nontrivial zeros of the Riemann zeta function naturally correspond to the vanishing ground state energies in this model. The model provides a natural form of supersymmetry.

Fractional derivatives of fuzzy complex-variable functions and application to fuzzy fractional complex differential equations

In this paper, through theory of fuzzy complex analysis, we introduce some representations of r-cuts of trivial fuzzy zero, nontrivial fuzzy zero and fuzzy complex numbers in polar form on complex plane and establish some of their basic properties. Based on this, the addition and scalar multiplication operations of fuzzy complex numbers are established and the calculus of fuzzy complex-variable functions is studied. Moreover, we introduce a notion of fuzzy Ortigueira fractional derivative of fuzzy complex-variable functions and develop a theory of fuzzy fractional calculus on complex plane including some geometric features. As applications, the existence and uniqueness theorem of solutions for nonlinear fuzzy fractional complex-variable differential equations is established and some examples are provided.

Topologically nontrivial and trivial zero modes in chiral molecules

Topologically nontrivial and trivial zero modes in chiral molecules

Origin Point Must Represent Critical Line as Location for Nontrivial Zeros of Riemann Zeta Function, and Set Prime Gaps With Subsets Odd Primes Are Arbitrarily Large in Number

We prove the Countably Infinite Subsets of odd primes have cardinality of Arbitrarily Large in Number. This is achieved by demonstrating the asymptotic law of distribution of prime numbers that involves natural logarithm function to be applicable to all these subsets of odd primes derived from every even Prime gaps which are again Arbitrarily Large in Number. We prove the location for Countably Infinite Set of nontrivial zeros computed using Dirichlet eta function (proxy function for Riemann zeta function) is the unique critical line. This is achieved by plotting the infinitely many colinear lines of Riemann zeta function using both critical line and non-critical lines.

New estimates for some integrals of functions defined over primes

In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the prime counting function $\pi(x)$ and the Chebyshev $\vartheta$-function. Some of these estimates depend on the correctness of the Riemann hypothesis on the nontrivial zeros of the Riemann zeta function $\zeta(s)$.

A simple note on the number-theoretic function Omega (n) and Dirichlet L-functions

We give a relation between the orders of potential non-trivial zeros of Dirichlet L-functions and analytic properties of certain Dirichlet series first introduced by Selberg.

On Ranks of Matrices over Noncommutative Domains

We consider matrices with entries in some domain, i.e., in a ring, not necessarily commutative, not containing non-trivial zero divisors. The concepts of the row rank and the column rank are discussed. (Coefficients of linear dependencies belong to the domain ; left coefficients are used for rows, right coefficients for columns.) Assuming that the domain satisfies the Ore conditions, i.e., the existence of non-zero left and right common multiples for arbitrary non-zero elements, it is proven that these row and column ranks are equal, which allows us to speak about the rank of a matrix without specifying which rank (row or column) is meant. In fact, the existence of non-zero left and right common multiples for arbitrary non-zero elements of is a necessary and sufficient condition for the equality of the row and column ranks of an arbitrary matrix over. An algorithm for calculating the rank of a given matrix is proposed. Our Maple implementation of this algorithm covers the domains of differential and (-)difference operators, both ordinary and with partial derivatives and differences.

The Quantized Constants with Remmen’s Scattering Amplitude to Explain Riemann Zeta Zeros

Riemann Hypothesis has been proposed by Bernhard Riemann since year 1859. Nowadays, there are lots of proof or disproof all over the internet society or the academic professional authority etc. However, none of them is accepted by the Clay’s Mathematics Institute for her Millennium Prize. In the past few months, this author discovered that there may be a correlation exists between the real and imaginary parts of Riemann Zeta function for the first 10 non-trivial zeros of the Riemann function etc. Indeed, when one tries to view the correlation relationship as a constant like the Planck’s one. Then we may show that Riemann Zeta zeros are indeed discrete quantum energy levels or the discrete spectrum as electrons falling from some bound quantum state to a lower energy state (or Quantum Field Theory). That may be further explained by Remmen’s scattering amplitude or the S-matrix. We may approximate the S-matrix by applying the HKLam theory to it and predict the scattering amplitude or even the Riemann Zeta non-trival zeros etc. By the way, the key researching equations or formula in the following content will be around the Taylor expansion of the Riemann Zeta function, their convergence etc. In additional, I will also investigate the (*’’) as shown below: ∏_(i=1)^∞▒(z-z_i ) = ξ(0.5 + i*t) = (∑_(n=1)^∞▒1)⁄n^((0.5+i*t) ) = ∏_(j=1)^∞▒(1-1⁄(p_j^((0.5+i*t) ) ))^(-1) ------------ (*’’) as we may find the existence of some constants like the Planck’s one. For the application of the aforementioned scholarly outcome, it is well-known that if one can find the pattern of the appearance to the prime number and hence break the public key cryptography in the everyday usage of information technology security etc.

Phenomenological formula for quantum Hall resistivity based on the Riemann zeta function

We propose a formula constructed out of elementary functions that captures many of the detailed features of the transverse resistivity ρ xy for the integer quantum Hall effect. It is merely a phenomenological formula in the sense that it is not based on any transport calculation for a specific class of physical models involving electrons in a disordered landscape, thus, whether a physical model exists which realizes this resistivity remains an open question. Nevertheless, since the formula involves the Riemann zeta function and its non-trivial zeros play a central role, it is amusing to consider the implications of the Riemann hypothesis in light of it.