Another 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 $\frac{1}{2}$. 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.

Proposed Proof of the Riemann Hypothesis

10.33774/coe-2021-h3m6b ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Prime Number ◽

Riemann Hypothesis ◽

Chebyshev Function

For every prime number $q_{n}$, we define the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$, where $\theta(x)$ is the Chebyshev function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. This is known as the Nicolas inequality. The Nicolas criterion states that the Riemann hypothesis is true if and only if the Nicolas inequality is satisfied for all primes $q_{n} > 2$. We prove indeed that the Nicolas inequality is satisfied for all primes $q_{n} > 2$. In this way, we show that the Riemann hypothesis is true.

The Riemann Hypothesis is True

10.33774/coe-2021-wd17h-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 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.

Possible Counterexample of the Riemann Hypothesis

10.33774/coe-2022-kxpg8-v2 ◽

2022 ◽

Author(s):

Frank Vega

Keyword(s):

Asymptotic Formula ◽

Real Number ◽

Riemann Hypothesis ◽

Chebyshev Function ◽

Precise Version

Under the assumption that the Riemann hypothesis is true, von Koch deduced the improved asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$, where $\theta(x)$ is the Chebyshev function. A precise version of this was given by Schoenfeld: He found under the assumption that the Riemann hypothesis is true that $\theta(x) < x + \frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x$ for every $x \geq 599$. On the contrary, we prove if there exists some real number $x \geq 2$ such that $\theta(x) > x + \frac{1}{\log \log \log x} \times \sqrt{x} \times \log^{2} x$, then the Riemann hypothesis should be false. In this way, we show that under the assumption that the Riemann hypothesis is true, then $\theta(x) < x + \frac{1}{\log \log \log x} \times \sqrt{x} \times \log^{2} x$.

The Nicolas criterion for the Riemann Hypothesis

10.33774/coe-2021-wfzw6-v4 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Prime Number ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex 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 $\frac{1}{2}$. 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.

Disproof of the Riemann Hypothesis

10.33774/coe-2021-0xvfk-v4 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Positive Constant ◽

Riemann Hypothesis ◽

A Value ◽

Large Numbers

We define the function $\upsilon(x)=\frac{3 \times \log x+5}{8 \times \pi \times \sqrt{x}+1.2 \times \log x+2}+\frac{\log x}{\log (x + C \times \sqrt{x} \times \log \log \log x)} - 1$ for some positive constant $C$ independent of $x$. We prove that the Riemann hypothesis is false when there exists some number $y \geq 13.1$ such that for all $x \geq y$ the inequality $\upsilon(x) \leq 0$ is always satisfied. We know that the function $\upsilon(x)$ is monotonically decreasing for all sufficiently large numbers $x \geq 13.1$. Hence, it is enough to find a value of $y \geq 13.1$ such that $\upsilon(y) \leq 0$ since for all $x \geq y$ we would have that $\upsilon(x) \leq \upsilon(y) \leq 0$. Using the tool $\textit{gp}$ from the project PARI/GP, we note that $\upsilon(100!) \approx \textit{-2.938735877055718770 E-39} < 0$ for all $C \geq \frac{1}{1000000!}$. In this way, we claim that the Riemann hypothesis could be false.

Another Criterion For The Riemann Hypothesis

10.20944/preprints202002.0379.v8 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Riemann Hypothesis ◽

Chebyshev Function

Let's define $\delta(x) = (\sum_{{q\leq x}}{\frac{1}{q}}-\log \log x-B)$, where $B \approx 0.2614972128$ is the Meissel-Mertens constant. The Robin theorem states that $\delta(x)$ changes sign infinitely often. For $x \geq 2$, Nicolas defined the function $u(x) = \sum_{q > x} \left(\log( \frac{q}{q-1}) - \frac{1}{q} \right)$ and proved that $0 < u(x) \leq \frac{1}{2 \times (x - 1)}$. We define the another function $\varpi(x) = \left(\sum_{{q\leq x}}{\frac{1}{q}}-\log \log \theta(x)-B \right)$, where $\theta(x)$ is the Chebyshev function. Using the Nicolas theorem, we demonstrate that the Riemann Hypothesis is true if and only if the inequality $\varpi(x) > u(x)$ is satisfied for all number $x \geq 3$. Consequently, we show that when the inequality $\varpi(x) \leq 0$ is satisfied for some number $x \geq 3$, then the Riemann Hypothesis should be false. Moreover, if the inequalities $\delta(x) \leq 0$ and $\theta(x) \geq x$ are satisfied for some number $x \geq 3$, then the Riemann Hypothesis should be false. In addition, we know that $\lim_{{x\to \infty }} \varpi(x) = 0$ because of $\lim_{{x\to \infty }} \delta(x) = 0$ and $\lim_{{x \to \infty }} \frac{\theta(x)}{x} = 1$.

Disproof of the Riemann Hypothesis

10.33774/coe-2021-0xvfk-v5 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Positive Constant ◽

Riemann Hypothesis ◽

A Value ◽

Large Numbers

We define the function $\upsilon(x)=\frac{3 \times \log x+5}{8 \times \pi \times \sqrt{x}+1.2 \times \log x+2}+\frac{\log x}{\log (x + C \times \sqrt{x} \times \log \log \log x)} - 1$ for some positive constant $C$ independent of $x$. We prove that the Riemann hypothesis is false when there exists some number $y \geq 13.1$ such that for all $x \geq y$ the inequality $\upsilon(x) \leq 0$ is always satisfied. We know that the function $\upsilon(x)$ is monotonically decreasing for all sufficiently large numbers $x \geq 13.1$. Hence, it is enough to find a value of $y \geq 13.1$ such that $\upsilon(y) \leq 0$ since for all $x \geq y$ we would have that $\upsilon(x) \leq \upsilon(y) \leq 0$. Using the tool $\textit{gp}$ from the project PARI/GP, we note that $\upsilon(100!) \approx \textit{-2.938735877055718770 E-39} < 0$ for all $C \geq \frac{1}{1000000!}$. In this way, we claim that the Riemann hypothesis could be false.