The Nicolas criterion for the Riemann Hypothesis

Riemann Zeta

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 $\frac{1}{2}$. For every prime number $p_{n}$, we define the sequence $X_{n} = \prod_{q \leq p_{n}} \frac{q}{q-1} - e^{\gamma} \times \log \theta(p_{n})$, where $\theta(x)$ is the Chebyshev 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 $X_{n} > 0$ holds for all primes $p_{n} > 2$. For every prime number $p_{k} > 2$, $X_{k} > 0$ is called the Nicolas inequality. We prove that the Nicolas inequality holds for all primes $p_{n} > 2$. In this way, we demonstrate that the Riemann hypothesis is true.

The Riemann Hypothesis is most likely true

10.33774/coe-2021-978nv ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex Numbers ◽

Natural Numbers ◽

Chebyshev Function ◽

Riemann Zeta ◽

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.

The Riemann Hypothesis Is True

10.33774/coe-2021-wd17h-v4 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex Numbers ◽

Natural Numbers ◽

Chebyshev Function ◽

Riemann Zeta ◽

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.

The Riemann Hypothesis Is True

10.33774/coe-2021-wd17h-v5 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex Numbers ◽

Natural Numbers ◽

Chebyshev Function ◽

Riemann Zeta ◽

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.

Arguments in Favor of the Riemann Hypothesis

10.33774/coe-2021-bpzxv ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex Numbers ◽

Natural Numbers ◽

Chebyshev Function ◽

Riemann Zeta ◽

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 $\gamma \approx0.57721$ 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 some arguments in favor of the Riemann hypothesis is true.

The Riemann Hypothesis Is Possibly True

10.33774/coe-2021-n8w1n ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex Numbers ◽

Natural Numbers ◽

Chebyshev Function ◽

Riemann Zeta ◽

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 possibly true.

A Proof to the Riemann Hypothesis

10.31219/osf.io/gbnpz ◽

2020 ◽

Author(s):

Sourangshu Ghosh

Keyword(s):

Number Theory ◽

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Prime Numbers ◽

Complex Numbers ◽

Riemann Zeta

In this paper, we shall try to prove the Riemann Hypothesis which is a conjecture that the Riemann zeta function hasits zeros only at the negative even integers and complex numbers with real part ½. This conjecture is very importantand of considerable interest in number theory because it tells us about the distribution of prime numbers along thereal line. This problem is one of the clay mathematics institute’s millennium problems and also comprises the 8ththe problem of Hilbert’s famous list of 23 unsolved problems. There have been many unsuccessful attempts in provingthe hypothesis. In this paper, we shall give proof to the Riemann Hypothesis.

When the Robin inequality does not hold

10.33774/coe-2021-w3582 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Prime Factor ◽

Correct Solution ◽

Complex Numbers ◽

Large N ◽

Riemann Zeta ◽

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 $\frac{1}{2}$. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality $\sigma(n)<e^{\gamma }\timesn\times\log\log n$ holds for all sufficiently large $n$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma\approx0.57721$ is the Euler-Mascheroni constant. In 1984, Guy Robin proved that the inequality is true for all $n>5040$ if and only if the Riemann Hypothesis is true. Let $n>5040$ be $n=r\timesq$, where $q$ denotes the largest prime factor of $n$. If $n>5040$ is the smallest number such that Robin inequality does not hold, then we show the following inequality is also satisfied: $\sqrt[q]{e}+\frac{\log\log r}{\log\log n}>2$.

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

Symmetry ◽

10.3390/sym13122410 ◽

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 ◽

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.

On the Riemann Hypothesis for the Zeta Function

10.20944/preprints202105.0616.v2 ◽

2021 ◽

Author(s):

Xiao-Jun Yang

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

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.