Zeta Zeros Research Articles

A Quantization for the Toy Model of Black Hole and a Suggestion to the Unification Theory

In my previous paper about the correlations between quantum mechanics and the four different natural forces, I suggested that there may be a fifth force or even the existence of a new particle. However, it remains a mystery to physicists, even after nearly a hundred years, that they cannot unify these four natural forces. This may be because there are indeed too many variables for them to consider. Also, there is a possibility that quantum mechanics may coexist with quantum field theory. In the present paper, this author proposes that there may be a misconception in the computational equation for relative time or gaps in the system for measuring time between quantum mechanics and general relativity. Hence, one may still not be able to unify these natural forces. This author suggests that we may need to rewrite parts of quantum mechanics – the Schrödinger equation – or even the general relativity equation. Additionally, this author proposes that there may be a bridge equation converting between quantum mechanics and general relativity. Moreover, this author has employed the non-trivial zeta zeros to simulate the black hole or the so-called black hole toy model. In practice, there may be an electromagnetic field surrounding the boundary of the black hole, as well as the existence of a continuum along the boundary contour. This author hopes that, in such a case, we may decode those high-frequency electromagnetic waves into useful information and take a further step toward verifying Stephen Hawking’s famous theory on black hole radiation and information entropy. In fact, this author has used the HKLam statistical model theory to express the electromagnetic field energy-stress tensor (with the possibility of quantization) to analogically establish a quantized model for the Einstein Gravitational Field Equation. Hence, the problem of quantum gravity may then be solved. Last but not least, this author also expects that humanity may finally find a way to unify quantum mechanics and general relativity through modifications to the current quantum gravity theory, such as my proposed bridge-converting equation, etc.

Zeta zeros and prolate wave operators

Zeta zeros and prolate wave operators

On the Approximation of the Hardy Z-Function via High-Order Sections

The Z-function is the real function given by Z(t)=eiθ(t)ζ12+it, where ζ(s) is the Riemann zeta function, and θ(t) is the Riemann–Siegel theta function. The function, central to the study of the Riemann hypothesis (RH), has traditionally posed significant computational challenges. This research addresses these challenges by exploring new methods for approximating Z(t) and its zeros. The sections of Z(t) are given by ZN(t):=∑k=1Ncos(θ(t)−ln(k)t)k for any N∈N. Classically, these sections approximate the Z-function via the Hardy–Littlewood approximate functional equation (AFE) Z(t)≈2ZN˜(t)(t) for N˜(t)=t2π. While historically important, the Hardy–Littlewood AFE does not sufficiently discern the RH and requires further evaluation of the Riemann–Siegel formula. An alternative, less common, is Z(t)≈ZN(t)(t) for N(t)=t2, which is Spira’s approximation using higher-order sections. Spira conjectured, based on experimental observations, that this approximation satisfies the RH in the sense that all of its zeros are real. We present a proof of Spira’s conjecture using a new approximate equation with exponentially decaying error, recently developed by us via new techniques of acceleration of series. This establishes that higher-order approximations do not need further Riemann–Siegel type corrections, as in the classical case, enabling new theoretical methods for studying the zeros of zeta beyond numerics.

A Quantization for the Toy Model of Black Hole and a Suggestion to the Unification Theory

In my previous paper about the correlations about the quantum mechanics between four different natural forces, I have suggested that there may be a fifth force or even the existence of a new particle. However, it may be still a mysterious that physicists for nearly hundreds of years that they cannot unify these four natural forces. This may be because there are indeed too many variables for them considerate. Also there is a twin such that the quantum machanics may coexist with the quantum field theory. In the present paper, this author proposes that there may be a misconcept in the computational equation for our relative time or the gaps in the system of measure time between quantum mechanics and general relativity. Hence, one may still cannot unify these natural forces. This author suggests that we need to rewrite parts of our QM – the Schrödinger Equation or even the GR equation. Besides, this author also proposes there may be a bridge converting equation for the quantum mechanics and the general relativity. Moreover, this author has employed the non-trivial zeta zeros to stimulate the black hole or the so-called black hole toy model. In practice, there may be an electromagnetic field laying around the boundary of the black hole as well as the existing of a continumm in the such boundary contour. This author hopes that in such case, we may decode back those high frequency electromagnetic waves into the useful information and may go a further step to have a verification for the famous Stephen Hawking’s theory between the black hole radition and the information entropy. In fact, this author has used the HKLam statistical model theory to express the electromagnetic field energy-stress tensor (with the possibility of quantization) to analogically establish a quantized model for the Einstein Gravitational Field Equation. Hence, the problem of the quantum gravity may then be solved. Last but not least, this author may also expect that we – human may finally find a way to unify the quantum mechanics and the general relativity through the modification to the present quantum gravity theory from my suggested bridge converting equation etc.

Negative discrete moments of the derivative of the Riemann zeta‐function

AbstractWe obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For , our bounds for the ‐th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros.

Arithmetic Consequences of the GUE Conjecture for Zeta Zeros

Arithmetic Consequences of the GUE Conjecture for Zeta Zeros

Smooth numbers and the Dickman ρ function

We establish an asymptotic formula for ψ(x, y) whose shape is xρ(log x/ log y) times correction factors. These factors take into account the contributions of zeta zeros and prime powers and the formula can be regarded as an (approximate) explicit formula for ψ(x, y). With this formula at hand we prove oscillation results for ψ(x, y), which resolve a question of Hildebrand on the range of validity of ψ(x, y) ≍ xρ(log x/ log y). We also address a question of Pomerance on the range of validity of ψ(x, y) ≥ xρ(log x/ log y).Along the way we improve classical estimates for ψ(x, y) and, on the Riemann Hypothesis, uncover an unexpected phase transition of ψ(x, y)at y = (log x)3/2+o(1).

Smooth integers and de Bruijn's approximation Ʌ

This paper is concerned with the relationship of $y$ -smooth integers and de Bruijn's approximation $\Lambda (x,\,y)$ . Under the Riemann hypothesis, Saias proved that the count of $y$ -smooth integers up to $x$ , $\Psi (x,\,y)$ , is asymptotic to $\Lambda (x,\,y)$ when $y \ge (\log x)^{2+\varepsilon }$ . We extend the range to $y \ge (\log x)^{3/2+\varepsilon }$ by introducing a correction factor that takes into account the contributions of zeta zeros and prime powers. We use this correction term to uncover a lower order term in the asymptotics of $\Psi (x,\,y)/\Lambda (x,\,y)$ . The term relates to the error term in the prime number theorem, and implies that large positive (resp. negative) values of $\sum _{n \le y} \Lambda (n)-y$ lead to large positive (resp. negative) values of $\Psi (x,\,y)-\Lambda (x,\,y)$ , and vice versa. Under the Linear Independence hypothesis, we show a Chebyshev's bias in $\Psi (x,\,y)-\Lambda (x,\,y)$ .

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.

Mean values of the logarithmic derivative of the Riemann zeta‐function near the critical line

AbstractAssuming the Riemann hypothesis and a hypothesis on small gaps between zeta zeros (see equation (ES 2K) below for a precise definition), we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith [J. Math. Phys. 60 (2019), no. 8, 083509], which states that for any positive integer K and real number , When , this was essentially a result of Goldston, Gonek and Montgomery [J. Reine Angew. Math. 537 (2001), 105–126] (see Equation (1) below).

On the number variance of zeta zeros and a conjecture of Berry

AbstractAssuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed non‐zero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non‐universal regime. In this range, Gaussian unitary ensemble statistics do not describe the distribution of the zeros. We also calculate lower order terms in the second moment of the logarithm of the modulus of the Riemann zeta function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000).

The boundary between the Prime Gaps and the Interchangeable be-tween a Quantum Mechanics System & a Classical Chaos One

With reference to my last paper, this author found that there are always some discrete quantum constants like the Planck’s one laying between the real and the imaginary parts of the Riemann function (I.e when we fix or group the Riemann Zeros one by one and make the real part to be varying.) or the quantum mechan-ics system. Hence, the Riemann Zeta Zeros give us the discrete quantum con-stants together with the given discrete quantum constants may give you the wanted Riemann Zeta Zeros. This phenomenon may be explained by Quantum Riemann Zeta Epiphancy. In the present paper, this author tries to fix or group the real part of the Riemann Zeros and make the imaginary part of the Riemann Zeros to be varying from one to the tenth one or resulting a chaos like graph. Thus, it seems that we may interchange a quantum mechanics system with the classical chaos one by just varying one variable and fix another one (such as varying real part of the Riemann function’s zeors from 0.1 to 1 and fix the imaginary part of the Rie-mann zeros one group after one another). Certainly, the vice versa for the fix of the real part (one group after another) with different Riemann Zeros may also help. In other words, the quantum mechanics system is just the conjugate pair (varying x and fixed y variables) of the classical chaos one or the vice versa is al-so true (fixed x and varying y variables) & put back into the Zeta functions etc. Hence, it is a method (or transformation) for us so that we may easily to make a interchange between these two systems by only adjusting or selecting which vari-ables to be varying or fixed. Then we may turn the classical chaos system into a quantum mechanics one or the vice versa. This implies that it may be possible to apply controllable quantum systems for the investigation of our natural aspects. Finally, this author have also shown that there is a boundary laying be-tween the prime gap through the elementary calculation and mathematical deriva-tion without using the computer software etc. It is thus a contradiction to the fact that there is no bound for the prime gap. This tells us that there is a need to have a shift from zero to 0.5 for the real part of the Riemann Zeta Zeros as what this author has mentioned in my last paper etc.

Small gaps and small spacings between zeta zeros

Small gaps and small spacings between zeta zeros

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

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

Distribution of Leading Digits of Imaginary Parts of Riemann Zeta Zeros

Abstract In this paper we study the distribution of leading digits of imaginary parts of Riemann zeta zeros in the b-adic expansion.

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.

The UV prolate spectrum matches the zeros of zeta

SignificanceWe show that the eigenvalues of the self-adjoint extension (introduced by A.C. in 1998) of the prolate spheroidal operator reproduce the UV behavior of the squares of zeros of the Riemann zeta function, and we construct an isospectral family of Dirac operators whose spectra have the same UV behavior as those zeros.

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.

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.