A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function

The completed zeta function $\xi(s)$ is expanded in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) of quadratic factors by its complex conjugate zeros $\rho_i=\alpha_i +j\beta_i, \bar{\rho}_i=\alpha_i-j\beta_i, 0<\alpha_i<1, \beta_i\neq 0, i\in \mathbb{N}$ are natural numbers from 1 to infinity, $\rho_i$ are in order of increasing $|\rho_i|=\sqrt{\alpha_i^2+\beta_i^2}$, i.e., $|\rho_1|<|\rho_2|\leq|\rho_3|\leq |\rho_4|, \cdots$, together with $\beta_1<\beta_2\leq\beta_3\leq\beta_4, \cdots$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\xi(0)\prod_{i\in \mathbb{N}}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)} =\xi(0)\prod_{i\in \mathbb{N}}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(1-s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}$$ which, by Lemma 3, is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}, \text{from 1 to infinity.}$$ with only valid solution $\alpha_i= \frac{1}{2}$ (another solution $s=\frac{1}{2}$ is invalid due to obvious contradiction). Thus, a proof of the Riemann Hypothesis is achieved.

Possible Counterexample of the Riemann Hypothesis

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 $\theta(x) < x + \frac{1}{\log \log \log x} \times \sqrt{x} \times \log^{2} x$.

Possible Counterexample of the Riemann Hypothesis

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 $\theta(x) > x + \frac{1}{\log \log x} \times \sqrt{x} \times \log^{2} x$, then the Riemann hypothesis should be false. In this way, we show that under the assumption that the Riemann hypothesis is true, then $\theta(x) < x + \frac{1}{\log \log x} \times \sqrt{x} \times \log^{2} x$.

Investigation of a Control Function Determining the Location of the Zeros as a Possible Proof of the Riemann Hypothesis.

The article examines the control function in relation to the distribution of Zeros on thecritical line x = 0,5. To confirm this hypothesis, it will be necessary to perform a large number ofstatistical analyzes of the distribution of non-trivial zero points of the Riemann Zeta function.

EULER PRODUCT ASYMPTOTICS FOR DIRICHLET -FUNCTIONS

The aim of this article is to establish the behaviour of partial Euler products for Dirichlet L-functions under the generalised Riemann hypothesis (GRH) via Ramanujan’s work. To understand the behaviour of Euler products on the critical line, we invoke the deep Riemann hypothesis (DRH). This work clarifies the relation between GRH and DRH.

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.

Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis

The Riemann Hypothesis states that the Riemann zeta function ζ(z) admits a set of “non-trivial” zeros that are complex numbers supposed to have real part 1/2. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. Hilbert and Pólya suggested that the Riemann Hypothesis could be solved through the mathematical tools of physics, finding a suitable Hermitian or unitary operator that describe classical or quantum systems, whose eigenvalues distribute like the zeros of ζ(z). A different approach is that of finding a correspondence between the distribution of the ζ(z) zeros and the poles of the scattering matrix S of a physical system. Our contribution is articulated in two parts: in the first we apply the infinite-components Majorana equation in a Rindler spacetime and compare the results with those obtained with a Dirac particle following the Hilbert-Pólya approach showing that the Majorana solution has a behavior similar to that of massless Dirac particles and finding a relationship between the zeros of zeta end the energy states. Then, we focus on the S-matrix approach describing the bosonic open string scattering for tachyonic states with the Majorana equation. Here we find that, thanks to the relationship between the angular momentum and energy/mass eigenvalues of the Majorana solution, one can explain the still unclear point for which the poles and zeros of the S-matrix of an ideal system that can satisfy the Riemann Hypothesis, exist always in pairs and are related via complex conjugation. As claimed in the literature, if this occurs and the claim is correct, then the Riemann Hypothesis could be in principle satisfied, tracing a route to a proof.