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

2021 ◽  
Author(s):  
Weicun Zhang
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Hypothesis ◽  
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 by conjugate complex roots. Then, according to Lemma 3, Lemma 4, and Lemma 5, the functional equation $\xi(s)=\xi(1-s)$ leads to $(s-\alpha_i)^2 = (1-s-\alpha_i)^2$ with solution $\alpha_i= \frac{1}{2}$, where $\alpha_i$ are the real parts of the zeros of $\xi(s)$, i.e., $s_i =\alpha_i\pm j\beta_i, i\in \mathbb{N}$. Thus a proof of the Riemann Hypothesis is achieved.

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.

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 ◽  
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 $\alpha_i\pm j\beta_i, i\in \mathbb{N}$. 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 is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$$ with solution $\alpha_i= \frac{1}{2}, i\in \mathbb{N}$. Therefore, 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

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 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 ◽  
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 $\alpha_i\pm j\beta_i, i\in \mathbb{N}$. 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

2022 ◽  
Author(s):  
Weicun Zhang
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Hypothesis ◽  
Hadamard Product ◽  
Infinite Product ◽  
Complex Conjugate ◽  
Maclaurin Series ◽  
Natural Numbers ◽  
Valid Solution

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 $\rho_i=\alpha_i +j\beta_i, \bar{\rho}_i=\alpha_i-j\beta_i, 0<\alpha_i<1, \beta_i\neq 0, i\in \mathbb{N}$ are natural numbers from 1 to infinity, $\rho_i$ are in order of increasing $|\rho_i|=\sqrt{\alpha_i^2+\beta_i^2}$, i.e., $|\rho_1|<|\rho_2|\leq|\rho_3|\leq |\rho_4|, \cdots$, together with $\beta_1<\beta_2\leq\beta_3\leq\beta_4, \cdots$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\xi(0)\prod_{i\in \mathbb{N}}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)} =\xi(0)\prod_{i\in \mathbb{N}}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(1-s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}$$ which, by Lemma 3, is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}, \text{from 1 to infinity.}$$ with only valid solution $\alpha_i= \frac{1}{2}$ (another solution $s=\frac{1}{2}$ is invalid due to obvious contradiction). Thus, a proof of the Riemann Hypothesis is 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 $\alpha_i\pm j\beta_i, \beta_i\neq 0, i\in \mathbb{N}$ are natural numbers, from $1$ to infinity, $\mathbb{N}$ is the set of natural numbers. 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 Ihara Zeta function, coefficients of Maclaurin series and Ramanujan graphs

2020 ◽  
Vol 31 (10) ◽  
pp. 2050082
Author(s):  
Hau-Wen Huang
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Finite Order ◽  
Undirected Graph ◽  
Number Sequence ◽  
Maclaurin Series ◽  
Ihara Zeta Function ◽  
Ramanujan Graphs ◽  
Weil Bound

Let [Formula: see text] denote a connected [Formula: see text]-regular undirected graph of finite order [Formula: see text]. The graph [Formula: see text] is called Ramanujan whenever [Formula: see text] for all nontrivial eigenvalues [Formula: see text] of [Formula: see text]. We consider the variant [Formula: see text] of the Ihara Zeta function [Formula: see text] of [Formula: see text] defined by [Formula: see text] The function [Formula: see text] satisfies the functional equation [Formula: see text]. Let [Formula: see text] denote the number sequence given by [Formula: see text] In this paper, we establish the equivalence of the following statements: (i) [Formula: see text] is Ramanujan; (ii) [Formula: see text] for all [Formula: see text]; (iii) [Formula: see text] for infinitely many even [Formula: see text]. Furthermore, we derive the Hasse–Weil bound for the Ramanujan graphs.

scholarly journals Some Remarks and Propositions on Riemann Hypothesis

2020 ◽  
Author(s):  
Jamal Salah
Keyword(s):  
Functional Equation ◽  
Integral Representation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Equal Zero ◽  
Riemann Zeta

In this article we look at some well know results of Riemann Zeta function in a different light. We explore the proofs of Zeta integral Representation, Analytic continuity and the first functional equation. We propound some modifications in order to reasonably justify the location of the non-trivial zeros on the critical line: s= 1/2 by assuming that ζ(s) and ζ(1-s) simultaneously equal zero

Sign in / Sign up

Export Citation Format

Share Document