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.
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 $$ξ(s)=(s-1)π^{-s/2}Γ(1+s/2)ζ(s).$$
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
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。
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 $…
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 …
