18 pages, replacement of the original claim that Riemann hypothesis has been proven with the statement that superconformal invariance implies Riemann hypothesis
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 coherent states and in general not orthogonal to each other. The states orthogonal to a vacuum state (having a negative norm squared) correspond to the zeros of Riemann Zeta. The physical states having a positive norm squared correspond to the zeros of Riemann Zeta at the critical line. Riemann hypothesis follows by reductio ad absurdum from the hypothesis that ordinary superconformal algebra acts as gauge symmetries for all coherent states orthogonal to the vacuum state, including also the non-physical coherent states that might exist off from the critical line.
The Riemann Hypothesis is not proved yet and this article gives a possible proof for the hypothesis which confirms that the only possible nontrivial zeros of the Riemann zeta-function has its real value equal to 1/2. From the result, the application of the Riemann Hypothesis will be certified, e.g. research on prime numbers.
A new equivalence to the Riemann Hypothesis by means of the Salem integral equation
Instituto de Matemáticas y Aplicaciones (IMAULL), Universidad de La Laguna（拉戈纳大学数学与应用研究所）；Departamento de Análisis Matemático, Universidad de La Laguna（拉戈纳大学数学分析系）
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 $$ξ(s)=(s-1)π^{-s/2}Γ(1+s/2)ζ(s).$$
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(…
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.
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 …
本文给出了Rimann猜想的证明。A. M. Odlyzko和H. Riile的《猜想的反证》，给出了Mertens猜想的反证，用于证明Riemann猜想。本文引入了由偶数决定的指数函数的新的有限级数。将多个自然数的个数与函数级数的个数进行比较，求出其数值。本文在解析数论中引入了“重叠的自然数”的新概念。此外，证明了一个定理，它给出了一个更准确的结果，“novivarious零点”比黎曼假设。
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.
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…
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, 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.
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 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 $n$.
