Nontrivial Zeros Research Articles

A Verification of Riemann Non-Trivial Zeros by Complex Analysis by Matlab™ Computation

With my most recent paper, I tried to prove the Riemann Hypothesis by catching out those contradictory parts of the non-trivial zeros. In the present paper, I will try to verify these known values of Riemann nontrivial zeros by first using U.S.A. Matlab coding with a list of well-organized complex analysis theories. At the same time, as the major core of my verification is just a mono-direction one (i.e. there may be a possibility of the missing non-trivial zeros although the residue value is zero), hence this author try to solve such problem by assuming that there are some other zeros existing between the two known zeros but the contradiction arises – as singularity implies the residue has a value with a multiple of 2πi. In addition, this author also apply the ingenious design (or a hybrid skill) with Feynman technique and Integration by parts to solve a special zeta function integral. Next, this author finds that one may consider those non-trivial zeros as a Fourier transform (or an impulse) between other normal complex numbers. The result is consistent with my previous papers in quantum physics [23], [25] for the electron jumps or reverse. Hence, we may get the (dirac) delta equation for Riemann Zeta. Then we may formulate our quantum circuit & computer. Finally, this author concludes all findings with an algorithm for searching, finer and checking the non-trivial zeros like below: Step 1: Use the computer software with some suitable program codes for an elementary search of feasible non-trivial zeta values among the closed real-complex plane interval – Method Matlab Simulation for searching zeta zeros; Step 2: Substitute back the values laying in the contour interval for zeta as found in Step 1 into the limit of ln(zeta(z))/((zeta'(z)) ) in order to adjust the answer in a finer and accurate way (just like the case of Newton’s method etc) with more decimal digitals – Method Ingenious Design for finer the zeta zero’s values; Step 3: Employ the Cauchy Residue Theorem for a check and hence confirm the previous found non-trivial zeta roots’ uniqueness without any zeta zeros laying in between the two consecutive zeta roots – Method Cauchy’s Residue for checking those already found zeta zeros.

The Generalized Riemann Hypothesis on elliptic complex fields

<abstract><p>In this paper, we will introduce a new algebraic system called the elliptic complex, and consider the distribution of zeros of the function $ L(s, \chi) $ in the corresponding complex plane. The key to this article is to discover the limiting case of the Generalized Riemann Hypothesis on elliptic complex fields and, taking a series of elliptic complex fields as variables, to study the ordinary properties of their distributions about the non-trivial zeros of $ L(s, \chi) $. It is on the basis of these considerations that we will draw the following conclusions. First, the zeros of the function $ L(s, \chi) $ on any two elliptic complex planes correspond one-to-one. Then, all non-trivial zeros of the $ L(s, \chi) $ function on each elliptic complex plane are distributed on the critical line $ \Re(s) = {1\over 2} $ due to the critical case of the Generalized Riemann Hypothesis. Ultimately we proved the Generalized Riemann Hypothesis.</p></abstract>

A Verification of Riemann Non-Trivial Zeros by Complex Analysis by Matlab™ Computation

A Verification of Riemann Non-Trivial Zeros by Complex Analysis by Matlab™ Computation

A zero density estimate and fractional imaginary parts of zeros for L-functions

AbstractWe prove an analogue of Selberg’s zero density estimate for $\zeta(s)$ that holds for any $\textrm{GL}_2$ L-function. We use this estimate to study the distribution of the vector of fractional parts of $\gamma\boldsymbol{\alpha}$ , where $\boldsymbol{\alpha}\in\mathbb{R}^n$ is fixed and $\gamma$ varies over the imaginary parts of the nontrivial zeros of a $\textrm{GL}_2$ L-function.

Measure for Chaotic Scattering Amplitudes.

We propose a novel measure of chaotic scattering amplitudes. It takes the form of a log-normal distribution function for the ratios r_{n}=δ_{n}/δ_{n+1} of (consecutive) spacings δ_{n} between two (consecutive) peaks of the scattering amplitude. We show that the same measure applies to the quantum mechanical scattering on a leaky torus as well as to the decay of highly excited string states into two tachyons. Quite remarkably, the r_{n} obey the same distribution that governs the nontrivial zeros of Riemann ζ function.

An asymptotic formula for the number of prime solutions for multivariate linear equations

In this paper, we study the multivariate linear equations with arbitrary positive integral coefficients. Under the Generalized Riemann Hypothesis, we obtained the asymptotic formula for the linear equations with more than five prime variables. This asymptotic formula is composed of three parts, that is, the first main term, the explicit second main term and the error term. Among them, the first main term is similar with the former one, the explicit second main term is relative to the non-trivial zeros of Dirichlet L-functions, and our error term improves the former one.

Scalar modular bootstrap and zeros of the Riemann zeta function

Using the technology of harmonic analysis, we derive a crossing equation that acts only on the scalar primary operators of any two-dimensional conformal field theory with U(1)c symmetry. From this crossing equation, we derive bounds on the scalar gap of all such theories. Rather remarkably, our crossing equation contains information about all nontrivial zeros of the Riemann zeta function. As a result, we rephrase the Riemann hypothesis purely as a statement about the asymptotic density of scalar operators in certain two-dimensional conformal field theories. We discuss generalizations to theories with only Virasoro symmetry.

Riemann hypothesis supports topological propulsion and faster than light travel through space time

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 (see the text accompanying Figure 6). This suggests the y-axis is literally infinite and that infinity equals zero. In this case, it is zero distance in time and space (again, see the text accompanying Figure 6). 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. As an introduction to this idea, I’ll provide background info from “Cosmos” magazine. Then I’ll move on to ideas which, today, may seem as fictional as Zefram Cochrane’s first warp-drive flight in 2063 (in the movie “Star Trek: First Contact”). But today’s science fiction is sometimes a non-technical preview of tomorrow’s science and technology. Three things are essential for the movement of both Cosmos’ curved-space robot and the propulsion-less (by known means) spaceship – shape changing, friction, and gravity. Future computers will take care of the first condition when they transform parallelograms into topological shapes. Friction is accounted for by deletion of the 3rd dimension (possible because of holographic-universe theory) plus topology’s single surfaces and self-intersections plus general relativity’s refraction of light by gravity. And the third requirement is satisfied by general relativity’s statement that gravity is the curvature of space-time. Topological propulsion also 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.

Fourier Interpolation with Zeros of Zeta and L-Functions

We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other L-functions. We establish a duality principle for Fourier interpolation bases in terms of certain kernels of general Dirichlet series with variable coefficients. Such kernels admit meromorphic continuation, with poles at a sequence dual to the sequence of frequencies of the Dirichlet series, and they satisfy a functional equation. Our construction of concrete bases relies on a strengthening of Knopp’s abundance principle for Dirichlet series with functional equations and a careful analysis of the associated Dirichlet series kernel, with coefficients arising from certain modular integrals for the theta group.

Notes on “On (IO,O)-fuzzy rough sets based on overlap functions”

Qiao investigated the properties and topological structures of (IO,O)-fuzzy rough sets, which extended the classical conjunction operator in rough approximation operator to an overlap function O. However, there are some faults in the characterizations of (IO,O)-fuzzy rough sets, such as wrong conclusions and strict condition, even if the overlap function O is assumed to be a continuous t-norm with no non-trivial zero divisors. This paper further discusses (IO,O)-fuzzy rough sets and rectifies those faults. Moreover, some examples are presented to show those faults.

An asymptotic expansion for a Lambert series associated to the symmetric square L-function

Hafner and Stopple proved a conjecture of Zagier that the inverse Mellin transform of the symmetric square [Formula: see text]-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function [Formula: see text]. Later, Chakraborty et al. extended this phenomenon for any Hecke eigenform over the full modular group. In this paper, we study an asymptotic expansion of the Lambert series [Formula: see text] where [Formula: see text] is the [Formula: see text]th Fourier coefficient of a Hecke eigenform [Formula: see text] of weight [Formula: see text] over the full modular group.

Eigenvalue Density, Li's Positivity, and the Critical Strip

We rewrite the zero-counting formula within the critical strip of the Riemann zeta function as a cumulative density distribution;this subsequently allows us to formally derive an integral expression for the Li coefficients associated with the Riemann Xi-function which, in particular, indicate that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We conjecture the validity of this and related expressions without the need for the Riemann Hypothesis and also offer a physical interpretation of the result and discuss the Hilbert-Polya approach.

Order of zeros of Dedekind zeta functions

Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field L L has infinitely many nontrivial zeros of multiplicity at least 2 if L L has a subfield K K for which L / K L/K is a nonabelian Galois extension. We also extend this to zeros of order 3 when G a l ( L / K ) Gal(L/K) has an irreducible representation of degree at least 3, as predicted by the Artin holomorphy conjecture.

A modular relation involving non-trivial zeros of the Dedekind zeta function, and the generalized Riemann hypothesis

We give a number field analogue of a result of Ramanujan, Hardy and Littlewood, thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function. We also provide a Riesz-type criterion for the Generalized Riemann Hypothesis for ζK(s). New elegant transformations are obtained when K is a quadratic extension, one of which involves the modified Bessel function of the second kind.

The Riemann hypothesis and tachyonic off-shell string scattering amplitudes

The study of the mathbf{4}-tachyon off-shell string scattering amplitude A_4 (s, t, u) , based on Witten’s open string field theory, reveals the existence of poles in the s-channel and associated to a continuum of complex “spins” J. The latter J belong to the Regge trajectories in the t, u channels which are defined by - J (t) = - 1 - { 1over 2 } t = beta (t)= { 1over 2 } + i lambda ; - J (u) = - 1 - { 1over 2 } u = gamma (u) = { 1over 2 } - i lambda , with lambda = real. These values of beta ( t ), gamma (u) given by { 1over 2 } pm i lambda , respectively, coincide precisely with the location of the critical line of nontrivial Riemann zeta zeros zeta (z_n = { 1over 2 } pm i lambda _n) = 0. It is argued that despite assigning angular momentum (spin) values J to the off-shell mass values of the external off-shell tachyons along their Regge trajectories is not physically meaningful, their net zero-spin value J ( k_1 ) + J (k_2) = J ( k_3 ) + J ( k_4 ) = 0 is physically meaningful because the on-shell tachyon exchanged in the s-channel has a physically well defined zero-spin. We proceed to prove that if there were nontrivial zeta zeros (violating the Riemann Hypothesis) outside the critical line Real~ z = 1/2 (but inside the critical strip) these putative zeros don't correspond to any poles of the mathbf{4}-tachyon off-shell string scattering amplitude A_4 (s, t, u) . We finalize with some concluding remarks on the zeros of sinh(z) given by z = 0 + i 2 pi n, continuous spins, non-commutative geometry and other relevant topics.

Determination of Gaussian Integer Zeroes of F(x,z)=2x4−z3

In this paper the zeroes of the polynomial F(x,z)=2x4−z3 in Gaussian integers Z[i] are determined, a problem equivalent to finding the solutions of the Diophatine equation x4+y4=z3 in Z[i], with a focus on the case x=y. We start by using an analytical method that examines the real and imaginary parts of the equation F(x,z)=0. This analysis sheds light on the general algebraic behavior of the polynomial F(x,z) itself and its zeroes. This in turn allows us a deeper understanding of the different cases and conditions that give rise to trivial and non-trivial solutions to F(x,z)=0, and those that lead to inconsistencies. This paper concludes with a general formulation of the solutions to F(x,z)=0 in Gaussian integers. Results obtained in this work show the existence of infinitely many non-trivial zeroes for F(x,z)=2x4−z3 under the general form x=(1+i)η3 and c=−2η4 for η∈Z[i].

A discrete mean-value theorem for the higher derivatives of the Riemann zeta function

We show that the nth derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for n odd/even, respectively. We show this by giving a full asymptotic expansion of these sums.

The Non-Trivial Zeros of The Riemann Zeta Function through Taylor Series Expansion and Incomplete Gamma Function

The Riemann zeta <img src=image/13425982_01.gif> function <img src=image/13425982_02.gif> is valid for all complex number <img src=image/13425982_03.gif>, for the line <img src=image/13425982_04.gif> = 1. Euler-Riemann found that the function equals zero for all negative even integers: −2, −4, −6, ... (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines <img src=image/13425982_04.gif> = 0 and <img src=image/13425982_04.gif> = 1. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line <img src=image/13425982_05.gif>. As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis. The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics. The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems. This function is useful in number theory for investigating the anomalous behavior of prime numbers. If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers. Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients. Despite the fact that there are around 10¹³, non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided. Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of <img src=image/13425982_06.gif>, and omitting the impact of an indeterminate form <img src=image/13425982_07.gif>, that appears in Riemann's approach. In this study, we also consider the simultaneous occurrence <img src=image/13425982_06.gif> but we adopt an element-wise approach of the Taylor series by expanding <img src=image/13425982_08.gif> for all <img src=image/13425982_09.gif> = 1, 2, 3, ... at the real parts of the non-trivial zeta zeros lying in the critical strip for <img src=image/13425982_10.gif> is a non-trivial zero of <img src=image/13425982_11.gif>, we first expand each term <img src=image/13425982_08.gif> at <img src=image/13425982_12.gif> then at <img src=image/13425982_13.gif>. Then in this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros<img src=image/13425982_06.gif>, on the critical strip by the means of different representations of Zeta function. Consequently, proves that Riemann Hypothesis is likely to be true.

On the Absence of Zeros of Riemann Zeta-Function Out of &amp;amp;#8476;(&amp;lt;i&amp;gt;z&amp;lt;/i&amp;gt;) = 1/2

This work shows, after a brief introduction to Riemann zeta function , the demonstration that all non-trivial zeros of this function lies on the so-called “critical line”,, the one Hardy demonstrated in his famous work that infinite countable zeros of the above function can be found on it. Thus, out of this strip, the only remaining zeros of this function are the so-called “trivial ones” . After an analytical introduction reminding the existence of a germ from a generic zero lying in , we show through a Weierstrass-Hadamard representation approach of the above germ that non-trivial zeros out of cannot be found.

General relativity from $p$-adic strings

For an arbitrary prime number $p$, we propose an action for bosonic $p$-adic strings in curved target spacetime, and show that the vacuum Einstein equations of the target are a consequence of worldsheet scaling symmetry of the quantum $p$-adic strings, similar to the ordinary bosonic strings case. It turns out that certain $p$-adic automorphic forms are the plane wave modes of the bosonic fields on $p$-adic strings, and that the regularized normalization of these modes on the $p$-adic worldsheet presents peculiar features which reduce part of the computations to familiar setups in quantum field theory, while also exhibiting some new features that make loop diagrams much simpler. Assuming a certain product relation, we also observe that the adelic spectrum of the bosonic string corresponds to the nontrivial zeros of the Riemann Zeta function.