Cotangent Sums Related to the Estermann Zeta Function and to the Riemann Hypothesis

2022 ◽  
pp. 65-144
Keyword(s):  
Zeta Function ◽  
Riemann Hypothesis

scholarly journals The Riemann Hypothesis is most likely true

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Chebyshev Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Sum Of Divisors

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems and it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is most likely true.

scholarly journals The Riemann Hypothesis Is True

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Chebyshev Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Sum Of Divisors

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. This problem has remained unsolved for many years. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n>5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1}>e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}>2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is true.

scholarly journals The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2410
Author(s):  
Janyarak Tongsomporn ◽  
Saeree Wananiyakul ◽  
Jörn Steuding
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Real Line ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Riemann’S Zeta Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

scholarly journals Farey Sequences and the Riemann Hypothesis

2016 ◽  
Author(s):  
Darrell Cox
Keyword(s):  
Zeta Function ◽  
Riemann Hypothesis ◽  
Analytic Number Theory ◽  
Square Root ◽  
Nontrivial Zeros ◽  
Landau Theorem ◽  
Farey Sequences ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Function Zeros

Relationships between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem are discussed. Whether a function similar to Chebyshev&rsquo;s second function is square-root close to a line having a slope different from 1 is discussed. The nontrivial zeros of the Riemann zeta function can be used to approximate many functions in analytic number theory. For example, it could be said that the nontrival zeta function zeros and the M&ouml;bius function generate in essence the same function - the Mertens function. A different approach is to start with a sequence that is analogous to the nontrivial zeros of the zeta function and follow the same procedure with both this sequence and the nontrivial zeros of the zeta function to generate in essence the same function. A procedure for generating such a function is given.

scholarly journals The Riemann Hypothesis Is True

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Chebyshev Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Sum Of Divisors

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. This problem has remained unsolved for many years. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n>5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1}>e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}>2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is true.

scholarly journals On the Riemann Hypothesis for the Zeta Function

2021 ◽  
Author(s):  
Xiao-Jun Yang
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
The Riemann Zeta Function ◽  
Equivalence Theorems ◽  
Riemann Zeta

In this paper we address some variants for the products of Hadamard and Patterson. We prove that all zeros of the Riemann $\Xi$--function are real. We also prove that the Riemann hypothesis is true. The equivalence theorems associated with the Riemann zeta--function are obtained in detail.

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.

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