A proof of the Riemann Hypothesis

Riemann Hypothesis: a GGC factorisation

A GGC (Generalized Gamma Convolution) representation of Riemann's Xi-function is constructed.

On the Riemann Hypothesis and its generalizations

A proof for the original Riemann hypothesis is proposed based on the infinite Hadamard product representation for the Riemann zeta function and later generalized to Dirichlet L-functions. The extension of the hypothesis to other functions is also discussed.

Some possible approaches to the Riemann Hypothesis via the Li/Keiper constants

In this paper we consider some possible approaches to the proof of the Riemann Hypothesis using the Li criterion.

A Geometric Proof of Riemann Hypothesis

Beginning from the formal resolution of Riemann Zeta function, by using the formula of inner product between two infinite-dimensional vectors in the complex space, the author proved the world's baffling problem -- Riemann hypothesis raised by German mathematician B. Riemann in 1859.

Proof of generalized Riemann hypothesis for Dedekind zetas and Dirichlet L-functions

A short proof of the generalized Riemann hypothesis (gRH in short) for zeta functions $ζ_{k}$ of algebraic number fields $k$ - based on the Hecke's proof of the functional equation for $ζ_{k}$ and the method of the proof of the Riemann hypothesis derived in [$M_{A}$] (algebraic proof of the Riemann hypothesis) is given. The generalized Riemann hypothesis for Dirichlet L-functions is an immediately consequence of (gRH) for $ζ_{k}$ and suitable product formula which connects the Dedekind zetas with L-functions.

One page proof of the Riemann hypothesis

We give a short Wiener measure proof of the Riemann hypothesis based on a surprising, unexpected and deep relation between the Riemann zeta $ζ(s)$ and the trivial zeta $ζ_{t}(s):=Im(s)(2Re(s)-1)$.

The Disproof of the Riemann Hypothesis

We show that there is a contradiction between the Riemann's Hypothesis and some form of the theorem on the universality of the zeta function.

黎曼假设的一个简单证明

A Simple Proof of the Riemann's Hypothesis

该研究提出仅需本科数学知识即可完成的黎曼假设（RH）简单证明。

On the Riemann Hypothesis

A proposed solution to the Riemann Hypothesis

A Geometric Proof of Generalized Riemann Hypothesis

Beginning from the resolution of Dirichlet L function, using the inner product formula of infinite-dimensional vectors in the complex space, the author proved the world's baffling problem--Generalized Riemann hypothesis.

黎曼假设的一个简单证明

A Simple Proof of the Riemann Hypothesis

本文通过分析黎曼ζ函数的非平凡零点，证明其全部位于临界线上，从而解决了黎曼假设。

通过萨利姆积分方程建立黎曼假设的新等价关系

A new equivalence to the Riemann Hypothesis by means of the Salem integral equation

本文通过萨利姆积分方程提出黎曼假设的新等价条件，探索数论中的核心问题。

迈向黎曼猜想的证明

Towards a proof of the Riemann Hypothesis

本文通过重新审视辐角原理，提出可能解决黎曼猜想的新思路，并寻求合作者共同推进。

A short Brownian motion proof of the Riemann hypothesis

We give a short probabilistic (a Brownian motion) proof of the Riemann hypothesis based on some surprising, unexpected and deep algebraic conjecture (MAC in short) concerning the relation between the Riemann zeta $ξ$ and a trivial zeta $ζ_{t}$. That algebraic conjecture was firstly discovered and formulated in [MA]

How to Verify the Riemann Hypothesis

It is proposed that the validity, or not, of the Riemann Hypothesis might be established on the basis of the integral $$\int\frac{ξ(2s)}{ξ(s)}ds$$ where $$ξ(s)=(s-1)π^{-s/2}Γ(1+s/2)ζ(s).$$

The Riemann hypothesis - an elementary analytic approach based on complex Laplace transform

An elementary analytic proof of the famous Riemann hypothesis is given. The main "accent" of the proof is a both using of the 2-dimensional double real and complex Laplace integral representations of the Green function $\mid z \mid^{-2}$.

New results based on Riemann hypothesis is tenable

Starting from the Euler's identity, the author improved Riemann's results, discovered the relationship between the Riemann Zeta function and the prime function, and obtained two new corollaries based on Riemann hypothesis is tenable. From these corollaries, the author found the relationship between the p_m, its subscript m and the t_m, and obtains a complete table of primes (less than 5000).

A Brief note on the Riemann hypothesis

We deal with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note.

From Bombieri's Mean Value Theorem to the Riemann Hypothesis

From Bombieri's mean value theorem one can deduce the prime number theorem being equivalent to the Riemann hypothesis and the least prime P(q) satisfying P(q)= O(q^2 [ln q]^32) in any arithmetic progressions with common difference q.

The Riemann hypothesis proved

The Riemann hypothesis is proved by quantum-extending the zeta Riemann function to a quantum mapping between quantum $1$-spheres with quantum algebra $A=\mathbb{C}$, in the sense of A. Prástaro \cite{PRAS01, PRAS02}. Algebraic topologic properties of quantum-complex manifolds and suitable bordism groups of morphisms in the category $\mathfrak{Q}_{\mathbb{C}}$ of quantum-complex manifolds are utilized.

与狄利克雷ζ函数相关的某些级数与广义黎曼假设

Some Series Related to Extended Riemann Hypothesis for Dedekind Zeta Functions

本文在广义黎曼假设下，推导了与狄利克雷ζ函数相关的级数闭式，并证明闭式成立则所有高零点位于临界线，从而获得黎曼假设的等价陈述。

Closed-form formula of Riemann zeta function and eta function for all non-zero given complex numbers via sums of powers of complex functions to disprove Riemann Hypothesis

An explicit identity of sums of powers of complex functions presented via this a closed-form formula of Riemann zeta function produced at any given non-zero complex numbers. The closed-form formula showed us Riemann zeta function has no unique solution for any given non-zero complex numbers which means Riemann zeta function is entirely divergent. Infinitely many zeros of Riemann zeta function produced unfortunately those zeros also gives us non-zero values of Riemann zeta function. Among those zeros some of them are zeros of Riemann hypothesis. The present paper also discussed on eta function(…

On an arithmetical approach to the Riemann hypothesis

In the paper, we first prove a sufficient condition for the Riemann hypothesis which involves the order of magnitude of the partial sum of the Liouville function. Then we show a formula which is curiously related to the proved sufficient condition.

A positive answer to the Riemann hypothesis: A new result predicting the location of zeros

In this paper, a positive answer to the Riemann hypothesis is given by using a new result that predict the exact location of zeros of the alternating zeta function on the critical strip.

黎曼猜想的证明

Proof of Riemann hypothesis

本文通过证明一个具有零点在开放临界带内的函数的凸性，成功证明了黎曼猜想。

A further step in the proof of Riemann hypothesis

Paper has been withdrawn due to an error in the basic argument that the states corresponding to the zeros of Riemann Zeta with Re[s]<1/2 allow a Fourier expansion in the basis provided by the states having Re[s]>= 1/2.

Towards a resolution of the Riemann hypothesis

This article contains work associated with a resolution of the Riemann hypothesis, following work by Taylor \cite{prt}, Lagarias and Suzuki \cite{lagandsuz} and Ki \cite{ki}, as well as Pustyl'nikov \cite{pust, pust2} and Keiper \cite{keiper}. Functions $ξ_+(s)$ and $ξ_-(s)$ are considered, for which it is known that all zeros lie on the critical line. The Riemann hypothesis itself pertains to the question of the location of the zeros of the sum and difference of $ξ_+(s)$ and $ξ_-(s)$, and this is investigated. An argument is developed which prima facie establishes the validity of the Riemann …

黎曼假设的一个证明

A Proof Of The Riemann Hypothesis

该研究聚焦黎曼假设证明问题，通过分析交错黎曼ζ函数的收敛性，结合Rouche定理、Bolzano-Weierstrass定理与反证法，完成了黎曼假设的证明工作。

Proof of the Riemann's hypothesis

The paper provides the proof of the Rimann's conjecture. The results of the works of A. M. Odlyzko and H. te Riile "Disproof of the Conjecture", which gives a disproof of the Mertens hypothesis, using to prove the Riemann's hypothesis. This paper introduces new finite series of exponential function, which is determined by the even number. The amount of multiple natural numbers is compared with the amount of the functional progression and their numerical value is faund. The paper introduces new concepts in analytic number theory as "natural numbers that overlap". Also, a theorem proved, which g…

Many-Body Fermions and Riemann Hypothesis

We study the algebraic structure of the eigenvalues of a Hamiltonian that corresponds to a many-body fermionic system. As the Hamiltonian is quadratic in fermion creation and/or annihilation operators, the system is exactly integrable and the complete single fermion excitation energy spectrum is constructed using the non-interacting fermions that are eigenstates of the quadratic matrix related to the system Hamiltonian. Connection to the Riemann Hypothesis is discussed.

Riemann Hypothesis may be proved by induction

The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At least one of these identities may be applied to prove the Riemann Hypothesis by induction. Additionally using this approach, the new series for Euler's constant gamma has been found.

黎曼猜想的证明

Proof of the Riemann Hypothesis

本文通过积分表示求解 zeta 函数实部，证明了黎曼猜想，即所有非平凡零点的实部为 1/2。

黎曼假设的证伪

Disproof of the Riemann Hypothesis

该研究通过展开黎曼Xi零函数的无穷乘积，证明临界线外存在至少一组四零点，结合Xi函数零点倒数和为常数的结论证伪黎曼假设，并给出临界线外零点求和公式的下界估计。

The Riemann Hypothesis

In the paper the well known Riemann Hypothesis is proven. The proof is based on uniform approximation of the zeta function discs of the critical strip placed to the right from the critical line.The basic moment is a use of a new mesure introduced in the infinite dimensional unite cube different from the Haar or product measures

A Conjecture Regarding the Riemann Hypothesis as Visualized by t strings

The introduction of strings into the study of the Riemann Hypothesis provides a visualization of the genesis of zeros for the Zeta function. The method is heuristic and when originally introduced suggested strong visual evidence for the truth of the Riemann Hypothesis. Some sort of organizing principle for strings with similar t values is observed and points towards a region between (1, 0) and (0, 0) on the abscissa, and within order unity along the ordinate. Progress in understanding these observations has been made by expanding the domain of sigma from the critical strip, [0, 1], to the half…

A focus on the Riemann's Hypothesis

Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin Academy of Mathematic. In that paper, he proposed that this function, called Riemann-zeta function takes values 0 on the complex plane when s=0.5+it. This hypothesis has great significance for the world of mathematics and physics. This solutions would lead to innumerable completions of theorems that rely upon its truth. Over a billion zeros of the function have b…

Hypothesis of Riemann is rejected by definition

Hypothesis of Riemann is rejected by definition, because ζ(s), where s zeros of ζ(s)=0, is not be equal by definition to the particular sum, which it assumes to be equal. R(s) = 1/2 holds only for the zeros of ζ(s) = 0 and for the zeros of certain related functions. However, it does not hold for certain special generalized functions of ζ(), such the Zeta Hurwitz functions and their sums.

Riemann Hypothesis的正确性

The rightness of the Riemann Hypothesis

本文利用若干函数性质研究Riemann zeta函数零点，通过假设零点不具$\frac{1}{2}+ib$形式导出矛盾，证明Riemann Hypothesis成立。

Solution to the Riemann Hypothesis from geometric analysis of component series functions in the functional equation of zeta

This paper presents a new approach towards the Riemann Hypothesis. On iterative expansion of integration term in functional equation of the Riemann zeta function we get sum of two series functions. At the `non-trivial' zeros of zeta function, value of the series is zero. Thus, Riemann hypothesis is false if that happens for an `s' off the line $\Re(s)=1/2$ ( the critical line). This series has two components $f(s)$ and $f(1-s)$. For the hypothesis to be false one component is additive inverse of the other. From geometric analysis of spiral geometry representing the component series functions $…

A Proof of the Riemann Hypothesis Through the Nicolas Inequality

A work by Nicolas has shown that if it can be proven that a certain inequality holds for all $n$, the Riemann hypothesis is true. This inequality is associated with the Mertens theorem, and hence the Euler totient at $\prod_{k=1}^n p_k$, where $n$ is any integer and $p_n$ is the $n$-th prime. We shall show that indeed the Nicolas inequality holds for all $n$.

Riemann hypothesis and super-conformal invariance

A strategy for proving (not a proof of, as was the first over-optimistic belief) the Riemann hypothesis is suggested. The vanishing of Riemann Zeta reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian operator D^+ having the zeros of Riemann Zeta as its eigenvalues. The construction of D^+ is inspired by the conviction that Riemann Zeta is associated with a physical system allowing superconformal transformations as its symmetries and second quantization in terms of the representations of superconformal algebra. The eigenfunctions of D^+ are analogous to the so called…

All Complex Zeros of the Riemann Zeta Function Are on the Critical Line: Two Proofs of the Riemann Hypothesis

I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeros of the Riemann Zeta Function is the critical line. The methods and results of this paper are based on well-known theorems on the number of zeros for complex value functions (Jensen, Titchmarsh, Rouche theorems), with the Riemann Mapping Theorem acting as a bridge between the Unit Disk on the complex plane and the critical strip. By primarily relying on well-known theorems of complex analysis our approach makes this paper …

A Brief Note on the Riemann hypothesis II

We have dealt with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and showed some evidence to indicate the hypothesis. We briefly review the essential points and we also define a finite ratio in the functional equation from divergent quantities in this note.

Quantum Knots and Riemann Hypothesis

In this paper we propose a quantum gauge system from which we construct generalized Wilson loops which will be as quantum knots. From quantum knots we give a classification table of knots where knots are one-to-one assigned with an integer such that prime knots are bijectively assigned with prime numbers and the prime number 2 corresponds to the trefoil knot. Then by considering the quantum knots as periodic orbits of the quantum system and by the identity of knots with integers and an approach which is similar to the quantum chaos approach of Berry and Keating we derive a trace formula which …

A Proof of Riemann Hypothesis

The meromorphic function $W(s)$ introduced in the Riemann-Zeta function $ζ(s) = W(s) ζ(1-s)$ maps the line of $s = 1/2 + it$ onto the unit circle in $W$-space. $|W(s)| = 0$ gives the trivial zeroes of the Riemann-Zeta function $ζ(s)$. In the range: $0 < |W(s)| \neq 1$, $ζ(s)$ does not have nontrivial zeroes. $|W(s)|=1$ is the necessary condition for the nontrivial zeros of the Riemann-Zeta function. Writing $s = σ+ it$, in the range: $0 \leq σ\leq 1$, but $σ\neq 1/2$, even if $|W(s)|=1$, the Riemann-Zeta function $ζ(s)$ is non-zero. Based on these arguments, the nontrivial zeros of the Riemann…

关于黎曼假说的新方法

On a new approach to the Riemann hypothesis

本文在黎曼假说不成立的前提下，研究了与黎曼ζ函数非平凡零相关的渐进行为，探讨了Dirichlet L函数残差与特定函数的关系及其对黎曼假说的启示。

Concerning Riemann Hypothesis

We present a quantum mechanical model which establishes the veracity of the Riemann hypothesis that the non-trivial zeros of the Riemann zeta-function lie on the critical line of $ζ(s)$.

Failed attempt to disproof the Riemann Hypothesis

In this paper we are going to describe the results of the computer experiment, which in principle can rule out the Riemann Hypothesis. We use the sequence $c_k$ appearing in the \BD criterion for the RH. Namely we calculate $c_{100000}$ with thousand digits of accuracy using two different formulas for $c_k$ with the aim to disproof the Riemann Hypothesis in the case these two numbers will differ. We found the discrepancy only on the 996 decimal place (accuracy of $10^{-996}$). The reported here experiment can be of interest for developers of Mathematica and PARI/GP.

Realization of the Riemann Hypothesis via Coupling Constant Spectrum

We present a Non-relativistic Quantum mechanical model, which exhibits the realization of Riemann Conjecture. The technique depends on exposing the $S$-wave Jost function at zero energy and in identifying it with the Riemann $ξ(s)$ function following a seminal paper of N. N. Khuri.

Further Comments on Realization of Riemann Hypothesis via Coupling Constant Spectrum

We invoke Carlson's theorem to justify and to confirm the results previously obtained on the validity of Riemann Hypothesis via the coupling constant spectrum of the zero energy S-wave Jost function a la N. N. Khuri, for the real, repulsive inverse-square potential in non-relativistic quantum mechanics in 3 dimensions.

A Simple Counterexample to Havil's "Reformulation" of the Riemann Hypothesis

This is an elementary note. It corrects a mistake in the reformulation of the Riemann Hypothesis in J. Havil's book Gamma: Exploring Euler's Constant.

On SUSY-QM, fractal strings and steps towards a proof of the Riemann hypothesis

We present, using spectral analysis, a possible way to prove the Riemann's hypothesis (RH) that the only zeroes of the Riemann zeta-function are of the form s=1/2+iλ_n. A supersymmetric quantum mechanical model is proposed as an alternative way to prove the Riemann's conjecture, inspired in the Hilbert-Polya proposal; it uses an inverse eigenvalue approach associated with a system of p-adic harmonic oscillators. An interpretation of the Riemann's fundamental relation Z(s) = Z(1-s) as a duality relation, from one fractal string L to another dual fractal string L' is proposed.