Nontrivial Zeros Research Articles

A further generalization of sums of higher derivatives of the Riemann zeta function

We obtain a full asymptotic for the sum of [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th derivative of the Riemann zeta function, [Formula: see text] is a positive real number, and [Formula: see text] denotes a nontrivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height [Formula: see text], as [Formula: see text]. We also specify what the asymptotic formula becomes when [Formula: see text] is a positive integer, highlighting the differences in the asymptotic expansions as [Formula: see text] changes its arithmetic nature.

Interlacing of zeros of odd period polynomials

The work of Conrey, Farmer and Imamoglu shows that all nontrivial zeros of the odd period polynomial of a Hecke eigenform of level one lie on the unit circle. We further show that nontrivial zeros of odd period polynomials of Hecke eigenforms of different weights share certain interlacing property.

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.

A discrete mean value of the Riemann zeta function

AbstractIn this work, we estimate the sum over the nontrivial zeros of the Riemann zeta function where is a complex number with and and are Dirichlet polynomials. Moreover, we estimate the discrete mean value above for higher derivatives where is replaced by for all . The formulae we obtain generalize a number of previous results in the literature. As an application, assuming the Riemann hypothesis, we obtain the lower bound which was previously known under the generalized Riemann hypothesis, only in the case .

Half-Isolated Zeros and Zero-Density Estimates

Abstract We introduce a new method to detect the zeros of the Riemann zeta function, which is sensitive to the vertical distribution of the zeros. This allows us to prove there are few “half-isolated” zeros. By combining this with classical methods, we improve the Ingham–Huxley zero-density estimate under the assumption that the non-trivial zeros of the zeta function are restricted to lie on a finite number of fixed vertical lines. This has new consequences for primes in short intervals under the same assumption.

A smooth version of Landau’s explicit formula

In this paper, we present a smooth version of Landau’s explicit formula for the von Mangoldt arithmetical function. By assuming the validity of the Riemann hypothesis, we show that in order to determine whether a natural number [Formula: see text] is a prime number, it is sufficient to know the location of a number of nontrivial zeros of the Riemann zeta function of order [Formula: see text]. Next we use Heisenberg’s inequality to support the conjecture that this number of zeros cannot be essentially diminished.

Multiplicity of nontrivial zeros of primitive 𝐿-functions via higher-level correlations

We give universal bounds on the fraction of nontrivial zeros having given multiplicity for L L -functions attached to a cuspidal automorphic representation of G L m / Q \mathrm {GL}_m/\mathbb {Q} . For this, we apply the higher-level correlation asymptotic of Hejhal, Rudnick, and Sarnak in conjunction with semidefinite programming bounds.

Remarks on the Connection of the Riemann Hypothesis to Self-Approximation

By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function ζ(s) by shifts ζ(s+iτ). In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating ζ(s) with all but at most countably many accuracies ε>0. Also, the analogue of an equivalent in terms of positive density in short intervals is discussed.

The Wiener Index of Prime Graph $PG(\mathbb{Z}_n)$

Let G = (V, E) be a simple graph and (R, +, ·) be a ring with zero element 0R. The Wiener index of G, denoted by W(G), is defined as the sum of distances of every vertex u and v, or half of the sum of all entries of its distance matrix. The prime graph of R, denoted by P G(R), is defined as a graph with V (P G(R)) = R such that uv ∈ E(P G(R)) if and only if uRv = {0R} or vRu = {0R}. In this article, we determine the Wiener index of P G(Zn) in some cases n by constructing its distance matrix. We partition the set Zn into three types of sets, namely zero sets, nontrivial zero divisor sets, and unit sets. There are two objectives to be achieved. Firstly,we revise the Wiener index formula of P G(Zn) for n = p2 and n = p3 for prime number p and we compare this results with the results carried out by previous researchers. Secondly, we determine the Wiener index formula of P G(Zn) for n = pq, n = p2q, n =p 2q2, and n = pqr for distinct prime numbers p, q, and r.

Solubility of additive forms of twice odd degree over totally ramified extensions of Q2$\mathbb {Q}_2$

AbstractWe prove that an additive form of degree , odd over any totally ramified extension of has a nontrivial zero if the number of variables satisfies .

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.

Hamiltonian for the Hilbert–Pólya conjecture

We introduce a Hamiltonian to address the Hilbert–Pólya conjecture. The eigenfunctions of the introduced Hamiltonian, subject to the Dirichlet boundary conditions on the positive half-line, vanish at the origin by the nontrivial zeros of the Riemann zeta function. Consequently, the eigenvalues are determined by these nontrivial Riemann zeros. If the Riemann hypothesis (RH) is true, the eigenvalues become real and represent the imaginary parts of the nontrivial zeros. Conversely, if the Hamiltonian is self-adjoint, or more generally, admits only real eigenvalues, then the RH follows. In our attempt to demonstrate the latter, we establish the existence of a similarity transformation of the introduced Hamiltonian that is self-adjoint on the domain specified by an appropriate boundary condition, which is satisfied by the eigenfunctions through the vanishing of the Riemann zeta function. Our result can be extended to a broader class of functions whose zeros lie on the critical line.

Zeros of higher derivatives of Riemann zeta function

In this article, we extend the result of Conrey [4, Theorem 2] to shorter intervals for higher-order derivatives of the zeta function. That is we study the mean value of the product of two finite order derivatives of the zeta function multiplied by a mollifier in short intervals. In this process, we obtain better mollifier length in some short intervals compared to the length of mollifier implied by Conrey's result. These finer studies allow us to refine the error term of some classical results of Levinson and Montgomery [13], Ki and Lee [11] on zero density estimates of ζ(k). Further, we showed that almost all non-trivial zeros of Matsumoto-Tanigawa's ηk-function cluster near the critical line.

A weighted one-level density of the non-trivial zeros of the Riemann zeta-function

A weighted one-level density of the non-trivial zeros of the Riemann zeta-function

Riemann zeros as quantized energies of scattering with impurities

We construct an integrable physical model of a single particle scattering with impurities spread on a circle. The S-matrices of the scattering with the impurities are such that the quantized energies of this system, coming from the Bethe Ansatz equations, correspond to the imaginary parts of the non-trivial zeros of the the Riemann ζ(s) function along the axis \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document} of the complex s-plane. A simple and natural generalization of the original scattering problem leads instead to Bethe Ansatz equations whose solutions are the non-trivial zeros of the Dirichlet L-functions again along the axis \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document}. The conjecture that all the non-trivial zeros of these functions are aligned along this axis of the complex s-plane is known as the Generalised Riemann Hypothesis (GRH). In the language of the scattering problem analysed in this paper the validity of the GRH is equivalent to the completeness of the Bethe Ansatz equations. Moreover the idea that the validity of the GRH requires both the duality equation (i.e. the mapping s → 1 – s) and the Euler product representation of the Dirichlet L-functions finds additional and novel support from the physical scattering model analysed in this paper. This is further illustrated by an explicit counterexample provided by the solutions of the Bethe Ansatz equations which employ the Davenport-Heilbronn function \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{D}\\left(s\\right)$$\\end{document}, i.e. a function whose completion satisfies the duality equation χ(s) = χ(1 – s) but that does not have an Euler product representation. In this case, even though there are infinitely many solutions of the Bethe Ansatz equations along the axis \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document}, there are also infinitely many pairs of solutions away from this axis and symmetrically placed with respect to it.

On the Sums over Inverse Powers of Zeros of the Hurwitz Zeta Function and Some Related Properties of These Zeros

Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of the analytical function to find the sums over inverse powers of zeros for the incomplete gamma and Riemann zeta functions, polygamma functions, and elliptical functions. Here, the same theorem is applied to study such sums for the zeros of the Hurwitz zeta function ζ(s,z), including the sum over the inverse first power of its appropriately defined non-trivial zeros. We also study some related properties of the Hurwitz zeta function zeros. In particular, we show that, for any natural N and small real ε, when z tends to n = 0, −1, −2… we can find at least N zeros of ζ(s,z) in the ε neighborhood of 0 for sufficiently small |z+n|, as well as one simple zero tending to 1, etc.

The uniform distribution modulo one of certain subsequences of ordinates of zeros of the zeta function

AbstractOn the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $1/2+i\gamma$ of the Riemann zeta function, we show that the sequence \begin{equation*}\Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad \frac{ \log\big(| \zeta^{(m_{\gamma })} (\frac12+ i{\gamma }) | / (\!\log{{\gamma }} )^{m_{\gamma }}\big)}{\sqrt{\frac12\log\log {\gamma }}} \in [a, b] \Bigg\},\end{equation*} where the ${\gamma }$ are arranged in increasing order, is uniformly distributed modulo one. Here a and b are real numbers with $a<b$ , and $m_\gamma$ denotes the multiplicity of the zero $1/2+i{\gamma }$ . The same result holds when the ${\gamma }$ ’s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers $\gamma (\!\log T)/2\pi$ with ${\gamma }\in \Gamma_{[a, b]}$ and $0<{\gamma }\leq T$ .

Remarks on a formula of Ramanujan

Assuming an averaged form of Mertens’ conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyse the finer structure of the terms in a well-known formula of Ramanujan.

Degenerate stability of the Caffarelli–Kohn–Nirenberg inequality along the Felli–Schneider curve

We show that the Caffarelli–Kohn–Nirenberg (CKN) inequality holds with a remainder term that is quartic in the distance to the set of optimizers for the full parameter range of the Felli–Schneider (FS) curve. The fourth power is best possible. This is due to the presence of non-trivial zero modes of the Hessian of the deficit functional along the FS-curve. Following an iterated Bianchi–Egnell strategy, the heart of our proof is verifying a ‘secondary non-degeneracy condition’. Our result completes the stability analysis for the CKN-inequality to leading order started by Wei and Wu. Moreover, it is the first instance of degenerate stability for non-constant optimizers and for a non-compact domain.

Beyond the Riemann Hypothesis [v2

The functional equation of real variable that Riemann used in his paper was subjected to elementary operations. And I obtained a lot of complex functional equations that the Riemann zeta function follows respectively. Here, functional equation transformations were the main methods for obtaining the complex functional equations. Half of those are equivalent to the complete symmetric functional equation that the Riemann Xi function follows, and one of those has an origin symmetry with correction terms. From the origin symmetric functional equation including correction terms, the representation containing the leading term of the zeta function for any complex number was obtained. And the Riemann hypothesis was proved by applying reduction to absurdity. Moreover the general representation containing the leading term of the zeta function for any odd number of 3 or more was also obtained. By suitably combining those functional equations, I observed a new explicit formula for the zeta function. The Riemann hypothesis was again proven using the deductive method. And two types of general representations for the zeta function for any odd number of either 3 or 7, or more, were also obtained from the explicit formula. In total, three types of general representations for the zeta function for any odd number of either 3 or 7, or more, were discovered. Conversely, I defined a new function, named the Chi function, for the left side of the origin symmetric functional equation that includes corrective terms. The Chi function is similar to the Riemann Xi function and exhibits origin symmetry. Furthermore, I defined a new function, the eta function, which is similar to the zeta function. The eta function’s pole and trivial zeros are the same as those of the zeta function. Furthermore, the Chi and eta functions have the same non-trivial zeros on the imaginary axis. And I proposed a generalized Riemann hypothesis for the eta function that states that all non-trivial zeros lie on the imaginary axis. Since I was able to discover the explicit formula for the eta function, the deductive method was used to prove the generalized Riemann hypothesis for the eta function. As you know, there are different types of transformations between the prime numbers and the non-trivial zeros of the zeta function. I discovered that there are comparable transformations between the prime numbers and the non-trivial zeros of the eta function. Based on the results of numerical experiments, I proposed some conjectures referring to relationships between the prime numbers and non-trivial zeros of the eta function.