scholarly journals A Direct Proof for Riemann Hypothesis Based on Jacobi Functional Equation and Schwarz Reflection Principle

2016 ◽  
Vol 06 (04) ◽  
pp. 193-200
Author(s):  
Xiang Liu ◽  
Rybachuk Ekaterina ◽  
Fasheng Liu
Keyword(s):  
Functional Equation ◽  
Riemann Hypothesis ◽  
Direct Proof ◽  
Reflection Principle ◽  
Schwarz Reflection Principle

scholarly journals Further properties of the Bergman spaces of slice regular functions

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Fabrizio Colombo ◽  
J. Oscar González-Cervantes ◽  
Irene Sabadini
Keyword(s):  
Unit Ball ◽  
Half Space ◽  
Half Plane ◽  
Bergman Kernel ◽  
Bergman Spaces ◽  
Reflection Principle ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Complex Half Plane ◽  
Schwarz Reflection Principle

AbstractWe continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paperwe mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitableweights are taken into account. Finally,we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane.

scholarly journals On the Schwarz reflection principle.

1953 ◽  
Vol 2 (2) ◽  
pp. 151-156 ◽  
Author(s):  
A. J. Lohwater
Keyword(s):  
Reflection Principle ◽  
Schwarz Reflection Principle

scholarly journals A Proof of the Riemann Hypothesis Based on A New Expression of the Completed Zeta Function

2021 ◽  
Author(s):  
Weicun Zhang
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Hypothesis ◽  
Hadamard Product ◽  
Infinite Product ◽  
Complex Conjugate ◽  
Maclaurin Series

The completed zeta function $\xi(s)$ is expanded in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) by its complex conjugate zeros $\alpha_i\pm j\beta_i, \beta_i\neq 0, i\in \mathbb{N}$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)} =\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}$$ which, by Lemma 3 and Corollary 1, is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$$ with solution $\alpha_i= \frac{1}{2}, i\in \mathbb{N}$. Thus, a proof of the Riemann Hypothesis can be achieved.

scholarly journals A Proof of the Riemann Hypothesis Based on A New Expression of the Completed Zeta Function

2021 ◽  
Author(s):  
Weicun Zhang
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Hypothesis ◽  
Hadamard Product ◽  
Infinite Product ◽  
Complex Conjugate ◽  
Maclaurin Series ◽  
Natural Numbers

The completed zeta function $\xi(s)$ is expanded in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) of quadratic factors by its complex conjugate zeros $\alpha_i\pm j\beta_i, \beta_i\neq 0, i\in \mathbb{N}$ ($\mathbb{N}$ is the set of natural numbers, from $1$ to infinity). Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)} =\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}$$ which, by Lemma 3 and Corollary 1, is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$$ with solution $\alpha_i= \frac{1}{2}, i\in \mathbb{N}$ (another solution $s=\frac{1}{2}$ is invalid due to obvious contradiction). Thus, a proof of the Riemann Hypothesis is achieved.

scholarly journals Minimal Surfaces with Only Horizontal Symmetries

ISRN Geometry ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Márcio Fabiano da Silva ◽  
Guillermo Antonio Lobos ◽  
Valério Ramos Batista
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Reflection Principle ◽  
Triply Periodic Minimal Surfaces ◽  
Triply Periodic ◽  
Horizontal Type ◽  
Straight Lines ◽  
Schwarz Reflection Principle

The Schwarz reflection principle states that a minimal surface S in ℝ3 is invariant under reflections in the plane of its principal geodesics and also invariant under 180°-rotations about its straight lines. We find new examples of embedded triply periodic minimal surfaces for which such symmetries are all of horizontal type.

scholarly journals A Proof of the Riemann Hypothesis Based on MacLaurin Expansion and Hadamard Product of the Completed Zeta Function

2021 ◽  
Author(s):  
Weicun Zhang
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Hypothesis ◽  
Hadamard Product ◽  
Infinite Product ◽  
Maclaurin Series ◽  
The Real ◽  
Complex Roots

The basic idea is to expand the completed zeta function $\xi(s)$ in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) by conjugate complex roots. Finally, the functional equation $\xi(s)=\xi(1-s)$ leads to $(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$ with solution $\alpha_i= \frac{1}{2}, i \in \mathbb{N}$, where $\alpha_i$ are the real parts of the zeros of $\xi(s)$, i.e., $s_i =\alpha_i\pm j\beta_i, i\in \mathbb{N}$. Therefore, a proof of the Riemann Hypothesis is achieved.

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