About RH – a Millennium Problem – a brief look at our solution for the Riemann Hypothesis

The Riemann Hypothesis (RH) proposed in 1859, one of the most important problem of modern mathematics is still considered unsolved.

“A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.”(www.claymath.org)

Do they really consider solutions outside the realm of number theory? Here is our.

With the solution (earlier versions in [3], [4] and the updated version from [9] ) provided for the Alcantara-Bode equivalent formulation of RH ([2]), we claim that the Riemann Hypothesis holds. Consider these pages a walk-through of the process for proving RH.

On this page we’ll present two connected equivalent formulations of RH, given by Beurling in 1955 and Alcantara-Bode in 1993, reducing RH to a problem solvable by applied math techniques. Follows the introduction of our solution based on a criteria for the injectivity of the linear operators on separable Hilbert spaces.
1. Injectivity Criteria: on next page there are the steps and ideas used for solving $%RH, the theorem of the injectivity of the linear operators and its application to$ the Alcantara-Bode’s equivalent formulation of RH.
2. Latest development: we introduced a generic theorem on injectivity for linear operators on separable Hilbert spaces and separated the analysis of the operator restrictions from the finite rank operator approximations on finite dimension subspaces whose union is dense in the separable Hilbert space.
3. Other considerations and comments.

Riemann Hypothesis (RH)

According to https://www.claymath.org/millennium-problems/riemann-hypothesis, the German mathematician G.F.B. Riemann (1826 – 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + …, $s = \sigma + it$ , called the Riemann Zeta function.

The Zeta function has zeros at each negative even integer s ∈ {-2, -4, -6, …} called ‘trivial zeros’. Other than trivial zeros, the function is non zero outside the vertical open strip 0 < σ < 1. The Riemann hypothesis asserts that all interesting solutions of the equation ζ(s) = 0 lie on a certain vertical straight line, σ = 1/2. This has been checked for the first 10,000,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.

Citing from https: //www.claymath.org/ collections/ riemanns-1859-manuscript/ : “Bernhard Riemann’s paper, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (On the number of primes less than a given quantity), was first published in the Monatsberichte der Berliner Akademie, in November 1859. Just six manuscript pages in length, it introduced radically new ideas to the study of prime numbers — ideas which led, in 1896, to independent proofs by Hadamard and de la Vallée Poussin of the prime number theorem. This theorem, first conjectured by Gauss when he was a young man, states that the number of primes less than x is asymptotic to x/log(x). Very roughly speaking, this means that the probability that a randomly chosen number of magnitude x is a prime is 1/log(x).”

“Fact: This is the only paper Riemann wrote on number theory, and as we all know today this changed the way we look at several topics in number theory and in a sense contributed the most towards development of the modern day subject of analytic number theory.” (citation from https://terrytao.wordpress.com/ 2021/02/12/ 246b-notes-4- the-riemann-zeta-function-and-the-prime-number-theorem/comment-page-1/#comment-681880)

The RH’s equivalent formulations:

a) Beurling’s equivalent formulation of RH

In 1955, Weyl presented the work of Beurling “A closure problem related to the Riemann zeta function” ([1]), in which the author proved the following equivalent formulation of RH:

The Riemann zeta-function is free from zeros in the half-plane $\sigma > 1/p, 1 < p < \infty$ , if and only if the linear manifold of functions $\quad f(x) = \sum_{\nu=1,n}c_\nu\rho(\frac{\theta_\nu}{x}), \quad 0 < \theta_\nu \leq 1, \quad n = 1, 2,...$ where the $c_\nu$ are constants such that $f(1+0) = \sum c_\nu\theta_\nu = 0$ and $\rho(y,x)$ is the fractional part of the ratio (y/x), is dense in the space $L^p(0,1).$

Remark: Beurling made an interesting observation ([1]): “any property of the Riemann zeta-function may be expressed in terms of some other property of the function $\rho(x)$ defined as the fractional part of the real number x”. During the proof of his theorem, Beurling pointed out the following identity involving $% the integral operator \quad L^2(0,1) having the kernel latex \quad \rho(\theta / x)$ the zeta-function $\zeta$ (s): for $\sigma > 0$

$(1) \qquad \qquad \qquad \int_0^1\rho(\theta/x)x^{s-1}dx = \theta/(s-1) - (\theta)^s\zeta(s)/s$ .

The left term in the equality could be written as $\quad (T_\rho x^{s-1})(\theta)$ by observing that $T_\rho$ is the Hilbert-Schmidt integral operator defined on $L^2(0,1)$ with the kernel function $\rho(\theta / x)$ . This integral operator has been used by Alcantara-Bode for proving his equivalent formulation of the Riemann Hypothesis.

At that time, (1955), Weyl and Beurling were colleagues at the Inst. for Advanced Study, Princeton, New Jersey.

b) Alcantara-Bode’s equivalent formulation of RH

In 1993, reformulating the theorem of Beurling for p=2, Alcantara-Bode proved ([2]):

RH holds if and only if the integral operator $T_\rho$ defined on $L^2(0,1)$ by

$(2) \qquad \qquad \qquad (T_\rho u)(y) = \int_0^1\rho(y,x) u(x)dx$

where $\rho$ is the fractional part function, is injective (equivalently, its null space contains only the element zero: $N_{T_\rho}$ = {0}).

According to Alcantara-Bode, his work was carried out during his visits to the I.M.P.A., Rio de Janeiro, Brazil; to Mathematisches Institut der Universitat Heidelberg, Germany; and while attending the X ELAM, held in Cordoba, Argentina.

Thus, as a consequence of the two equivalent formulations to Riemann’s Hypothesis (RH), those of Beurling and of Alcantara-Bode, in order to prove RH is enough to prove the injectivity of the specific linear Hilbert-Schmidt integral operator, defined in (2) on the separable Hilbert space $L^2(0,1)$ .

The solution

For the investigation of the linear operators injectivity on the separable Hilbert spaces, we introduced the Injectivity Criteria theorem ([3], [4]), succinctly presented below.

Let $L_F$ be the class of bounded linear operators on a separable Hilbert space H, which are strict positive definite on every subspace of a dense family F of including finite dimension subspaces, and let $\mu_n$ be the injectivity parameter of the operator restriction $T_{\mid_{S_n}}$ , to the subspace $S_n \in F$ . The Injectivity Criteria theorem backing up the numerical method states:

let $T \in L_F$ . If there exists a constant C > 0 such that $\mu_n$ > C for every n ≥ 1, then $T$ is injective.

When the operator T is Hermitian, the injectivity parameter $\mu_n$ is defined as the ratio between the minimum and the maximum eigenvalues of $T_{\mid_{S_n}}$ .

An expression for the ratio $\mu_n$ is provided for non Hermitian operators too, and, that has been the context in which we proved that $N_{T_\rho}$ = {0}. This is a result equivalent with the proof of the RH as a consequence due to the Alcantara-Bode hypothesis.

Conclusion $(^*)$ , $(^+)$ : our solution to RH consisted in proving the injectivity of the integral operator defined in (2). It has been obtained by applying the Injectivity Criteria to the integral operator $T_\rho$ using a dense family of including subspaces in $L^2$ (0,1), family built on interval indicator functions whose density is well known in literature.

On the next page we will present the steps taken for obtaining this result.

The notes below are parts of the update of the method regarding the investigation of the injectivity (or of null space) of an Hilbert-Schmidt integral operator on a separable Hilbert space.

$(^*)$ We introduced in [8] a new theorem extending the Injectivity Criteria ([3], [4]): “Theorem 1: A linear bounded operator strict positive definite on a dense set of a separable Hilbert space, is injective”, simplifying the investigation of the linear operators injectivity.

$(^+)$ In [9] we used the compactness of the integral operator in order to obtain the result: “an Hilbert-Schmidt integral operator on a separable Hilbert space having the finite rank approximations with the sequence of the positivity parameters inferior bounded, is injective”. In other words, if there exists a constant $\alpha$ strict positive valued such that $\alpha_h \geq \alpha$ for $n\geq 2, nh=1$ , then $T_\rho$ is strict positive on the dense set, i.e. for every u from dense set $\langle T_\rho u, u\rangle \geq \alpha(T_\rho) \|u\|^2$ . Then from Theorem 1, $N_{T_\rho} = \{0\}$ .

The value of the positivity parameter is $\alpha(T_\rho) = \alpha - \epsilon_0$ , where $\epsilon_0$ is the closest constant verifying the relation $\epsilon_h$ < $\alpha$ , where $\epsilon_h := \|T_\rho - T_{\rho_h}\|$ . We remind that, from the compactness of $T_\rho$ the sequence $\epsilon_h$ converges to 0 with n converging to $\infty, nh=1.$

Dumitru Adam (dumitru_adam@yahoo.ca)

Notes : _{Our solution to RH has been published in [3] and [4] , reminding that [9] is an updated and complete solution fixing some flaws found in [8].}

_{All articles are open-access, distributed under the Creative Commons Attribution Non-Commercial License (CC BY-NC) which permits reuse, distribution and reproduction of the article, provided that the original work is properly cited and the reuse is restricted to noncommercial purposes .}

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Bibliography

[1] Beurling, A.(1955) “A closure problem related to the Riemann zeta function”, $\emph{Proc. Nat. Acad. Sci. 41}$ pg. 312-314, 1955.

[2] Alcantara-Bode, J. (1993) “An Integral Equation Formulation of the Riemann Hypothesis”, $\emph{Integr Equat Oper Th, Vol. 17}$ , 1993.

[3] Adam D., “On the Injectivity of an Integral Operator Connected to Riemann Hypothesis”, J. Pure Appl Math. 2022; 6(4):19-23 , https://www.pulsus.com/scholarly-articles/on-the-injectivity-of-an-integral-operator-connected-to-riemann-hypothesis.pdf (DOI: 10.37532/2752-8081.22.6(4).19-23)

[4] Adam D., “On the Injectivity of an Integral Operator Connected to Riemann Hypothesis”, Research Square https://doi.org/10.21203/rs.3.rs-1159792/v9 (crossref) – v1 dated Dec 2021

[5] Buescu, J., Paixao A. C., (2007) ”Eigenvalue distribution of Mercer-like kernels”, Math.Nachr.280,No.9–10, pg. 984 – 995, 2007.

[6] Chang, C.H., Ha, C.W. (1999) ”On eigenvalues of differentiable positive definite kernels”, Integr. Equ. Oper. Theory 33 pg. 1-7, 1999

[7] Adam, D. (1994) “Mesh Independence of Galerkin Approach by Preconditioning”, Preconditioned Iterative Methods – Johns Hopkins Libraries, Lausanne, Switzerland; [Langhome, Pa.] : Gordon and Breach, ©1994

[8] Adam, D., (2023) “An updated method on the injectivity of integral operators related to Riemann Hypothesis”, https://doi.org/10.21203/rs.3.rs-3777446/v5, preprint Research Square.

[9] Adam, D., (2024) “On the Method for Proving the RH Using the Alcantara-Bode Equivalence”, https://doi.org/10.20944/preprints202411.1062.v9, Preprints.org (www.preprints.org)