Riemann Hypothesis Research Articles

Class numbers, Ono invariants and some interesting primes

Our aim is to find all the prime numbers p such that p+x2 has at most two different prime factors, for all the odd integers x such that x2≤p. We solve entirely the cases p≡1,3,5(mod8), using the knowledge of the quadratic imaginary number fields with class numbers 4, 1 and 2 respectively. The case p≡7(mod8) is not completely solved. Taking into account a result of Stéphane Louboutin, we prove that there is at most one value p≡7(mod8) besides our list. Assuming a Restricted Riemann Hypothesis, the list is complete. In the last section of the paper we give a short sketch for the general problem: find all odd integers n>1 such that n+x2 has at most two different prime factors, for all the odd integers x such that x2≤n.

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.

Ratios conjecture for quadratic twists of modular L-functions

We develop the L-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic twist of modular L-functions using multiple Dirichlet series under the generalized Riemann hypothesis.

The Extension of the Riemann’s Zeta Function

In mathematics, the search for exact formulas giving all the prime numbers, certain families of prime numbers or the n-th prime number has generally proved to be vain, which has led to contenting oneself with approximate formulas [8]. The purpose of this article is to give a new proof of the Riemann hypothesis [4]-which is closely related to the distribution of prime numbers- by y introducing S^ a new extension of the of the Riemann zeta function.

Riemann hypothesis for period polynomials for cusp forms on Γ0(N)

We prove that for even integer k, almost all of zeros of the period polynomial associated to a cusp form of weight k on [Formula: see text] are on the circle [Formula: see text] under some conditions.

Properties of a Linear Operator Involving Lambert Series and Rabotnov Function

This work is an attempt to apply Lambert series in the theory of univalent functions. We first consider the Hadamard product of Rabotnov function and Lambert series with coefficients derived from the arithmetic function σn to introduce a normalized linear operator JRα,βz. We then acquire sufficient conditions for JRα,βz to be univalent, starlike and convex, respectively. Furthermore, we discuss the inclusion results in some special classes, namely, spiral-like and convex spiral-like subclasses. In addition, we extend the findings by incorporating two Robin’s inequalities, one of which is analogous to the Riemann hypothesis.

Quantitative upper bounds related to an isogeny criterion for elliptic curves

AbstractFor and elliptic curves defined over a number field , without complex multiplication, we consider the function counting nonzero prime ideals of the ring of integers of , of good reduction for and , of norm at most , and for which the Frobenius fields and are equal. Motivated by an isogeny criterion of Kulkarni, Patankar, and Rajan, which states that and are not potentially isogenous if and only if , we investigate the growth in of . We prove that if and are not potentially isogenous, then there exist positive constants , , and such that the following bounds hold: (i) ; (ii) under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) under GRH, Artin's Holomorphy Conjecture for the Artin ‐functions of number field extensions, and a Pair Correlation Conjecture for the zeros of the Artin ‐functions of number field extensions.

Moments of Derivatives of Modular L-Functions

Abstract Let f be a newform for the group $\Gamma_{0}(q)$ and χd be a primitive quadratic character of conductor $|d|$. In this article, we prove an asymptotic for the second moment of the derivative of $L(s,\, f \otimes \chi_{8d})$ at the central point $1/2$, which was previously known under grand Riemann hypothesis (GRH) by Petrow [9].

A new class of solutions to the van Dantzig problem, the Lee–Yang property, and the Riemann hypothesis

A new class of solutions to the van Dantzig problem, the Lee–Yang property, and the Riemann hypothesis

Large deviations of the argument of the Riemann zeta function

AbstractLet . We prove an unconditional lower bound on the measure of the sets for . For , our bound has a Gaussian shape with variance proportional to . At the endpoint, , our result implies the best known ‐theorem for that is due to Tsang. We also explain how the method breaks down for given our current knowledge about the zeros of the zeta function. Conditionally on the Riemann hypothesis, we extend our results to the range .

Robin inequality for n/phi(n)

Let $\varphi(n)$ be the Euler function, $\sigma(n)=\sum_{d\mid n}d$ the sum of divisors function and $\gamma=0.577\ldots$ the Euler constant. In 1982, Robin proved that, under the Riemann hypothesis, $\sigma(n)/n < e^\gamma \log\log n$ holds for $n > 5040$ and that this inequality is equivalent to the Riemann hypothesis. The aim of this paper is to give a similar equivalence for $n/\varphi(n)$.

BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET CHARACTER SUMS

BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET CHARACTER SUMS

Supersymmetry Via EDM (Electric Dipole Moment)

The article begins by mentioning the accepted correlation between Albert Einstein’s relativistic equations, as well as James Clerk Maxwell’s electromagnetic waves, with the Prandtl-Glauert equations for fluid flow. This equation-free mention nevertheless ends with 180° ΔO = + => - (180 Degree Change in Orientation Equals Positive Becomes Negative). Of course, it also means negative can become positive: 180° ΔO = - => +. Using this article’s new equations, the Electric Dipole Moment is introduced and the charges of the quarks composing a neutron are transformed. EDM’s undetectability is explained from the perspective of a hypothetical someone who doesn’t accept the Big Bang but believes the universe is literally infinite and eternal. Then quantum spin of matter particles is discussed, and extended to astronomy’s black holes, in terms of photons and gravitons not being elementary force-carrying particles but being in possession of EDM. The photon-graviton interaction producing quantum spin is proposed as electromagnetic and gravitational vectors in a figure that might be called “Vector-Tensor-Scalar Geometry”, as well as being related to quaternions plus the Higgs boson and field. Then the article returns to black holes, showing how the inability of light to escape them leads to a 4-dimensional space-time: via binary digits or base 2 mathematics, Mobius bands, figure-8 Klein bottles, and Wick rotation. The electric potential of photons and gravitons is then interpreted not strictly as a neutron-identical Electric Dipole Moment but as a vast array of pulses sharing similarities with modern computers and electronics – the binary digits of the previous sentence. As a consequence of the electric force-carriers bringing them into existence, particles of matter and antimatter are symmetrical with bosons in the sense that every boson or fermion is, at its most fundamental level, composed of binary digits. Imaginary numbers are essential to quantum mechanics, and this article connects the imaginary numbers of Wick rotation to the nature of time. Therefore, the words here are not painting a classical picture of space-time. The 1’s and 0’s of binary digits are compatible with quantum mechanics and may be referred to as the Hidden Variables which Einstein advocated to complete quantum physics, and to give its calculations an exactness which would bring a hidden order to its chaotic randomness and superficial uncertainty. If the universe can be quantized and viewed as comprised of infinitesimal ones and zeros, how could it not obey quantum physics? And if those ones and zeros are all ultimately connected by Quantum Gravity, waves and particles could never be separated but wave-particle duality would rule. To finish, many thanks to Albert Einstein for laying the foundations of this article 105 years ago when he published a paper titled “Do gravitational fields play an essential role in the structure of elementary particles?” Immediately before submission for publication, a few topics were added to this article - dark matter, precession, the Hodge conjecture, and the Riemann hypothesis.

An explicit version of Chen’s theorem assuming the Generalized Riemann Hypothesis

An explicit version of Chen’s theorem assuming the Generalized Riemann Hypothesis

On the Universal Presence of Mathematics for Incompletely Predictable Problems in Rigorous Proofs for Riemann Hypothesis, Modified Polignac’s and Twin Prime Conjectures

We validly ignore even prime number 2. Based on all arbitrarily large number of even Prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with the Prime number theorem for Arithmetic Progressions, and our derived Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. With these conditions being satisfied by all Odd Primes, we argue Polignac's and Twin prime conjectures are proven to be true when they are usefully treated as Incompletely Predictable Problems. In so doing [and with Riemann hypothesis being a special case], this action also support the generalized Riemann hypothesis formulated for Dirichlet L-function. By broadly applying Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture to Dirichlet eta function (proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true when it is usefully treated as Incompletely Predictable Problem. Crucial connections exist between these Incompletely Predictable Problems and Quantum field theory whereby eligible (sub-)functions and (sub-)algorithms are treated as infinite series.

A note on r$r$‐gaps between zeros of the Riemann zeta‐function

AbstractIn this paper, we prove Selberg's announced result on ‐gaps between zeros of the Riemann zeta‐function . Our proof uses a result on variations of by Tsang based on Selberg's method. The same result with explicit constants under the Riemann Hypothesis has been obtained by Conrey and Turnage‐Butterbaugh using a different method. We explain how to obtain explicit constants under the Riemann Hypothesis using our approach which is based on Selberg's and Tsang's arguments.

A note on the two variable Artin's conjecture

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b, the set{p⩽x:p prime, mamodp∈〈b〉modp} has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs (a,b)#{p⩽x:p prime, mamodp∈〈b〉modp}≫xlog2⁡x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S={m1,m2,m3} such thatm1,m2,m3,−3m1m2,−3m2m3,−3m1m3,m1m2m3 are not squares, there exists a pair of elements a,b∈S such that#{p⩽x:p prime, mamodp∈〈b〉modp}≫xlog⁡log⁡xlog2⁡x. Further, under the assumption of a level of distribution greater than x23 in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers S={m1,m2} such thatm1,m2,−3m1m2 are not squares, there exists a pair of elements a,b∈{m1,m2,−3m1m2} such that#{p⩽x:p prime, mamodp∈〈b〉modp}≫xlog⁡log⁡xlog2⁡x.

Discrete mean values of the product of derivatives of shifted Dirichlet L-functions

Assuming the Generalized Riemann Hypothesis, we calculate ∑0<γχ≤TL(μ)(ρχ+iδ,χ)L(ν)(1−ρχ−iδ,χ‾), which extends the result proposed by Gonek in 1984.

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.

Towards an elementary formulation of the Riemann hypothesis in terms of permutation groups

Towards an elementary formulation of the Riemann hypothesis in terms of permutation groups