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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 considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. In 2011, Solé and Planat stated that, the Riemann hypothesis is true if and only if the inequality $\zeta(2) \cdot \prod_{p\leq x} (1+\frac{1}{p}) > e^{\gamma} \cdot \log \theta(x)$ holds for all $x \geq 5$, where $\theta(x)$ is the first Chebyshev function, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant, $\zeta(x)$ is the Riemann zeta function and $\log$ is the natural logarithm. In this note, using Solé and Planat criterion, we prove that, when the Riemann hypothesis is false, then there are infinitely many natural numbers $x$ for which $\frac{\log x}{\sqrt{x}} - \frac{10}{\sqrt{x}} + 2 \cdot \log x + \varepsilon \cdot \log x \leq 2.062$ could be satisfied for some $\varepsilon > 0$. Since the inequality $\frac{\log x}{\sqrt{x}} - \frac{10}{\sqrt{x}} + 2 \cdot \log x + \varepsilon \cdot \log x \leq 2.062$ never holds for every $\varepsilon > 0$ and large enough $x$, then the Riemann hypothesis is true by principle of non-contradiction.

Under the assumption that the Riemann Hypothesis is true, von Koch deduced the asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$. A precise version of this was given by Schoenfeld. He found under the assumption that the Riemann Hypothesis is true that $\left| \theta(x) - x \right| < \frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x$ for every $x \geq 599$. We prove that if the Riemann Hypothesis is true, then \[\log \theta(x) > \log x - \log \left(\frac{8 \times \pi \times \sqrt{x}}{8 \times \pi \times \sqrt{x} - \log^{2} x}\right) - \left(\frac{4+\gamma -\log 4 \times \pi}{\sqrt{x-\frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x}}\right)\] for every $x \geq p_{120569}$, where $\gamma \approx 0.57721$ is the Euler-Mascheroni constant and $p_{120569}$ is the 120569th prime number.

Under the assumption that the Riemann hypothesis is true, von Koch deduced the asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$. A precise version of this was given by Schoenfeld. He found under the assumption that the Riemann hypothesis is true that $\left| \theta(x) - x \right| < \frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x$ for every $x \geq 599$. Using this result, we prove that if the Riemann hypothesis is true, then $\prod_{q \leq x} \frac{q}{q-1} > \left(e^{\gamma} \times \log x\right) \times \left(1 + \frac{\log (1 - \frac{1}{8 \times \pi \times \sqrt{x}} \times \log^{2} x)}{\log x}\right)$ for every $x \geq 599$.

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 $\frac{1}{2}$. In 2011, Sol{\'e} and and Planat stated that the Riemann Hypothesis is true if and only if the inequality $\prod_{q\leq q_{n}}\left(1+\frac{1}{q} \right) >\frac{e^{\gamma}}{\zeta(2)}\times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 3$, where $\theta(x)$ is the Chebyshev function, $\gamma\approx 0.57721$ is the Euler-Mascheroni constant and $\zeta(x)$ is the Riemann zeta function. Using this result, we create a new criterion for the Riemann Hypothesis. We prove the Riemann Hypothesis is true using this new criterion., Last version of the author without the final revision of the referees.

Under the assumption that the Riemann hypothesis is true, von Koch deduced the asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$. A precise version of this was given by Schoenfeld. He found under the assumption that the Riemann hypothesis is true that $\left| \theta(x) - x \right| < \frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x$ for every $x \geq 599$. Using this result, we prove that if the Riemann hypothesis is true, then $\prod_{q \leq x} \frac{q}{q-1} < \left(e^{\gamma} \times \log x\right) \times \left(1 - \frac{\log x}{8 \times \pi \times \sqrt{x}}\right)$ for every $x \geq 599$. Hence, we obtain that if the Riemann hypothesis is true, then $x^{\left(1 - \frac{\log x}{8 \times \pi \times \sqrt{x}}\right)} > \theta(x)$ for every $x \geq 599$. However, this is false since $(\theta(x) - x)$ changes sign infinitely often. By contraposition, the Riemann hypothesis is false.

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. On the contrary, we prove if there exists some real number $x \geq 10^{8}$ 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. Note that, the von Koch asymptotic formula uses the Big $O$ notation, where $f(x) = O(g(x))$ means that there exists a positive real number $M$ and a real number $y$, such that $|f(x)| \leq M \times g(x)$ for all $x \geq y$. However, no matter how big we get the real number $y \geq 10^{8}$, the another positive real number $M$ could always prevail over the value of $\frac{1}{\log \log \log x}$ for sufficiently large numbers $x \geq y$.

The current research regarding the Riemann zeros suggests the existence of a non-trivial algebraic/analytic structure on the set of Riemann zeros. The duality between primes and Riemann zeta function zeros suggests some new goals and aspects to be studied: {\em adelic duality} and the {\em POSet of prime numbers}. The article presents computational evidence of the structure of the imaginary parts $t$ of the non-trivial zeros of the Riemann zeta function $\rho=1/2+it$, called in this article the {\em Riemann Spectrum}, using the study of their distribution. The novelty represents in considering the associated characters $p^{it}$, towards an algebraic point of view, than rather in the sense of Analytic Number Theory. This structure is tentatively interpreted in terms of adelic characters, and the duality of the rationals. Second, the POSet structure of prime numbers studied, is tentatively mirrored via duality in the Riemann spectrum. A direct study of the convergence of their Fourier series, along Pratt trees, is proposed. Further considerations, relating the Riemann Spectrum, adelic characters and distributions, in terms of Hecke idelic characters, local zeta integrals (Mellin transform) and $\omega$-eigen-distributions, are explored following., Comment: 15 pages

Srinivasa Ramanujan studied the function $S_{1}(x) = \sum_{\rho} \frac{x^{\rho - 1}}{\rho \cdot (1 - \rho)}$ where $\rho$ runs over the nontrivial zeros of the Riemann $\zeta$ function. Under the Riemann hypothesis, we know that $\lvert S_{1}(x) \rvert \leq \frac{\tau}{\sqrt{x}}$ for $\tau = 2 + \gamma - \log(4 \cdot \pi) \approx 0.04619$. 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 considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. In 2011, Solé and Planat stated that, the Riemann hypothesis is true if and only if the inequality $\zeta(2) \cdot \prod_{p\leq x} (1+\frac{1}{p}) > e^{\gamma} \cdot \log \theta(x)$ holds for all $x \geq 5$, where $\theta(x)$ is the Chebyshev function, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant, $\zeta(x)$ is the Riemann zeta function and $\log$ is the natural logarithm. In this note, using Solé and Planat criterion, we prove that, when the Riemann hypothesis is false, then there are infinitely many natural numbers $x$ for which $\frac{\log x}{\sqrt{x}} - \frac{10}{\sqrt{x}} + 2 \cdot \log x + S_{1}(x) \cdot \sqrt{x} \cdot \log x \leq 2.062$ could be satisfied. In addition, we show that the Riemann hypothesis is true when $S_{1}(x) \geq \frac{\varepsilon}{\sqrt{x}}$ for $\varepsilon \geq -1.9999999$ and large enough $x$.

A trustworthy proof for the Riemann hypothesis has been considered as the Holy Grail of Mathematics by several authors. 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 $\frac{1}{2}$. Let $Q$ be the set of prime numbers $q_{n}$ satisfying the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1}>e^{\gamma} \cdot\log\theta(q_{n})$ with the product extending over all prime numbers $q$ that are less than or equal to $q_{n}$, where $\gamma\approx 0.57721$ is the Euler-Mascheroni constant, $\theta(x)$ is the Chebyshev function and $\log$ is the natural logarithm. If the Riemann hypothesis is false, then there are infinitely many prime numbers $q_{n}$ outside and inside of $Q$. In this note, we obtain a contradiction when we assume that there are infinitely many prime numbers $q_{n}$ outside of $Q$. By reductio ad absurdum, we prove that 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 1/2. In 2011, Patrick Solé and Michel Planat stated a new criterion for the Riemann Hypothesis. We prove the Riemann Hypothesis is true using this criterion.

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 theorem states that the Riemann hypothesis is true if and only if the $X_{n} > 0$ holds for all prime $p_{n} > 2$. For every prime number $p_{k}$, $X_{k} > 0$ is called the Nicolas inequality. We show if the sequence $X_{n}$ is strictly decreasing for $n$ big enough, then the Riemann hypothesis must be true. For every prime number $p_{n} > 2$, we define the sequence $Y_{n} = \frac{e^{\frac{1}{2 \times \log(p_{n})}}}{(1 - \frac{1}{\log(p_{n})})}$ and show that $Y_{n}$ is strictly decreasing for $p_{n} > 2$. For all prime $p_{n} > 286$, we demonstrate that the inequality $X_{n} < e^{\gamma} \times \log Y_{n}$ is always satisfied. We prove that $\lim_{{n\to \infty }}X_{n}=\lim_{{n\to \infty }}(\log Y_{n})=0$.

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 some arguments in favor of the Riemann hypothesis is true.

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. Let's also define $S(x) = \theta(x) - x$, where $\theta(x)$ is the Chebyshev function. It is known that $S(x)$ changes sign infinitely often. We define the another function $\varpi(x) = \left(\sum_{{q\leq x}}{\frac{1}{q}}-\log \log \theta(x)-B \right)$. We prove that when the inequality $\varpi(x) \leq 0$ is satisfied for some number $x \geq 3$, then the Riemann hypothesis should be false. The Riemann hypothesis is also false when the inequalities $\delta(x) \leq 0$ and $S(x)\geq 0$ are satisfied for some number $x \geq 3$ or when $\frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log \theta(x)} \leq 1$ is satisfied for some number $x \geq 13.1$ or when there exists some number $y \geq 13.1$ such that for all $x \geq y$ the inequality $\frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log (x + C \times \sqrt{x} \times \log \log \log x)} \leq 1$ is always satisfied for some positive constant $C$ independent of $x$.

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 found the first zero $y$ of the function $\upsilon(y)$ in $y \approx 8.2639316883312400623766461031726662911 \ E5565708$ for $C \geq 1$. In this way, we claim that the Riemann hypothesis could be false.

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.

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.

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 $\frac{1}{2}$. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all sufficiently large $n$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ 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. In 2002, Lagarias proved that if the inequality $\sigma(n) \leq H_{n} + exp(H_{n}) \times \log H_{n}$ holds for all $n \geq 1$, then the Riemann Hypothesis is true, where $H_{n}$ is the $n^{th}$ harmonic number. We show certain properties of these both inequalities that leave us to a verified proof of the Riemann Hypothesis. These results are supported by the claim that a numerical computer calculation verifies that the subtraction of \[\log (e^{\gamma} \times q_{m} \times r) + e^{\gamma} \times q_{m} \times r \times \log \log (e^{\gamma} \times q_{m} \times r)\] with \[(q_{m} + 1) \times \log (e^{\gamma} \times (r + 1)) + (q_{m} + 1) \times e^{\gamma} \times (r + 1) \times \log \log (e^{\gamma} \times (r + 1))\] is monotonically increasing as much as $q_{m}$ and $r$ become larger just starting with the initial values of $q_{m} = 47$ and $r = 1$, where $q_{m}$ is a prime number and $r$ is a natural number. In this way, we can confirm that the Riemann Hypothesis is true based on computational mathematics using a simple and naive computer assisted proof.

Let us denote by \(\pi (x)\) the number of primes \(\leqslant x\), by \({{\mathrm{li}}}(x)\) the logarithmic integral of x, by \(\theta (x)=\sum _{p\leqslant x} \log p\) the Chebyshev function and let us set \(A(x)={{\mathrm{li}}}(\theta (x))-\pi (x)\). Revisiting a result of Ramanujan, we prove that the assertion “\(A(x) > 0\) for \(x\geqslant 11\)” is equivalent to the Riemann Hypothesis.

Suppose that the Riemann hypothesis holds. Suppose that $$\psi _1 (x) = \mathop \sum \limits_{\begin{array}{*{20}c} {n \leqslant x} \\ {\{ (1/2)n^{1/c} \} < 1/2} \\ \end{array} } \Lambda (n)$$ where c is a real number, 1 N 1/2+10e, e > 0, the following asymptotic formula is valid: $$\psi _1 (N + H) - \psi _1 (N) = \frac{H}{2}\left( {1 + O\left( {\frac{1}{{N^\varepsilon }}} \right)} \right)$$ .

Suppose that the Riemann hypothesis is valid, and let ψ be the Chebyshev function. In this paper, we obtain the following bound for all β >1 and positive integers \(n \geqslant 2:\int_1^{T^\beta } {|\psi } (x + \frac{x}{T}) - \psi (x) - \frac{x}{T}|^n \frac{{{\text{d}}x}}{{x^{n/2 + 1} }} \ll _{n,\beta } \frac{{\log ^n T}}{T}, T \to \infty\).

This paper presents well-conditioned rational Chebyshev approximations, involving at most one exponentiation, for computation of either ζ ( s ) \zeta (s) or ζ ( s ) − 1 , .5 ≦ s ≦ 55 \zeta (s) - 1,.5 \leqq s \leqq 55 , for up to 20 significant figures. The logarithmic error is required in one case. An algorithm for the Hurwitz zeta function, and an example of nearly double degeneracy are also given.

The function $\epsilon(x)=\mbox{li}(x)-\pi(x)$ is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions $\epsilon_{\theta}(x)=\mbox{li}[\theta(x)]-\pi(x)$ and $\epsilon_{\psi}(x)=\mbox{li}[\psi(x)]-\pi(x)$ are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $\theta(x)=\sum_{p \le x} \log p$ and $\psi(x)=\sum_{n=1}^x \Lambda(n)$, respectively, $\mbox{li}(x)$ is the logarithmic integral, $\mu(n)$ and $\Lambda(n)$ are the M\"obius and the Von Mangoldt functions). Negative jumps in the above functions $\epsilon$, $\epsilon_{\theta}$ and $\epsilon_{\psi}$ may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One denotes $j_p=\mbox{li}(p)-\mbox{li}(p-1)$ and one investigates the jumps $j_p$, $j_{\theta(p)}$ and $j_{\psi(p)}$. In particular, $j_p1$ for $p, Comment: 15 pages section 2.2 added, new sequences added, Fig. 2 and 3 are new