Short Note on the Riemann Hypothesis

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 $\gamma\approx0.57721$ is the Euler-Mascheroni constant. Let $q_{1}=2,q_{2}=3,\ldots,q_{m}$ denote the first $m$ consecutive primes, then an integer of the form $\prod_{i=1}^{m}q_{i}^{a_{i}}$ with $a_{1} \geq a_{2} \geq \cdots \geq a_{m} \geq 0$ is called an Hardy-Ramanujan integer. If the Riemann hypothesis is false, then there are infinitely many Hardy-Ramanujan integers $n>5040$ such that Robin inequality does not hold and we prove that $n^{\left(1-\frac{0.6253}{\log q_{m}}\right)}<N_{m}$, where $N_{m}=\prod_{i =1}^{m}q_{i}$ is the primorial number of order $m$ and $q_{m}$ is the largest prime divisor of $n$. In addition, we show that $q_{m}$ will not have an upper bound by some positive value for these counterexamples and therefore, the value of $q_{m}$ tends to infinity as $n$ goes to infinity.

Short Note on the Riemann Hypothesis

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 $\gamma\approx0.57721$ is the Euler-Mascheroni constant. Let $q_{1}=2,q_{2}=3,\ldots,q_{m}$ denote the first $m$ consecutive primes, then an integer of the form $\prod_{i=1}^{m}q_{i}^{a_{i}}$ with $a_{1} \geq a_{2} \geq \cdots \geq a_{m} \geq 0$ is called an Hardy-Ramanujan integer. If the Riemann hypothesis is false, then there are infinitely many Hardy-Ramanujan integers $n>5040$ such that Robin inequality does not hold and we prove that $n^{\left(1-\frac{0.6253}{\log q_{m}}\right)}<N_{m}$, where $N_{m}=\prod_{i =1}^{m}q_{i}$ is the primorial number of order $m$ and $q_{m}$ is the largest prime divisor of $n$. In addition, we show that $q_{m}$ will not have an upper bound by some positive value for these counterexamples and therefore, the value of $q_{m}$ tends to infinity as $n$ goes to infinity.

Some new semiring structures

In this paper, we introduce some new semiring structures by considering the prime factors, sum of divisors and the number of relatively prime positive divisors of a pair of non-negative integers and solve a variety of system of linear equations over one of these semirings. We then discuss the properties of these structures and how they compare with classical algebra. The paper is concluded by providing a key-exchange scheme using one of the newly discovered semirings.

Proof of the Riemann Hypothesis

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

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 True

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

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 Possibly True

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.

The Riemann Hypothesis is most likely 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 is most likely 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.