The Riemann Hypothesis is True

The Riemann hypothesis has been considered the most important unsolved problem in pure mathematics. The David Hilbert's list of 23 unsolved problems contains the Riemann hypothesis. Besides, 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 Could Be True

10.33774/coe-2021-wd17h ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Riemann Hypothesis ◽

Unsolved Problem ◽

Natural Numbers ◽

Chebyshev Function ◽

Pure Mathematics ◽

The Riemann hypothesis has been considered to be the most important unsolved problem in pure mathematics. The David Hilbert's list of 23 unsolved problems contains the Riemann hypothesis. Besides, 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 could be true.

The Riemann Hypothesis is True

10.33774/coe-2021-wd17h-v2 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Riemann Hypothesis ◽

Unsolved Problem ◽

Natural Numbers ◽

Chebyshev Function ◽

Pure Mathematics ◽

The Riemann hypothesis has been considered the most important unsolved problem in pure mathematics. The David Hilbert's list of 23 unsolved problems contains the Riemann hypothesis. Besides, 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 true.

The Riemann Hypothesis is most likely true

10.33774/coe-2021-978nv-v3 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Riemann Hypothesis ◽

Unsolved Problem ◽

Natural Numbers ◽

Chebyshev Function ◽

Pure Mathematics ◽

The Riemann hypothesis has been considered the most important unsolved problem in pure mathematics. The David Hilbert's list of 23 unsolved problems contains the Riemann hypothesis. Besides, 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 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.

Proof of the Riemann Hypothesis

10.33774/coe-2021-s7270 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Riemann Hypothesis ◽

Unsolved Problem ◽

Natural Numbers ◽

The Riemann hypothesis has been considered the most important unsolved problem in mathematics. 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 $\sigma(n)$ is the sum-of-divisors function of $n$ and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We show that the Robin inequality is true for all natural numbers $n > 5040$ which are not divisible by the prime $3$. Moreover, we prove that the Robin inequality is true for all natural numbers $n > 5040$ which are divisible by the prime $3$. Consequently, the Robin inequality is true for all natural numbers $n > 5040$ and thus, 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.