The Complete Proof of the Riemann Hypothesis

Journal de Théorie des Nombres de Bordeaux, 2007

On Robin's criterion for the Riemann hypothesis par YoungJu CHOIE, Nicolas LICHIARDOPOL, Pieter MOREE et Patrick SOLÉ Résumé. Le critère de Robin spécifie que l'hypothèse de Riemann (RH) est vraie si et seulement si l'inégalité de Robin σ(n) := d|n d < e γ n log log n est vérifiée pour n ≥ 5041, avec γ la constante d'Euler(-Mascheroni). Nous montrons par des méthodeś elémentaires que si n ≥ 37 ne satisfait pas au critère de Robin il doitêtre pair et il n'est ni sans facteur carré ni non divisible exactement par un premier. Utilisant une borne de Rosser et Schoenfeld, nous montrons, en outre, que n doitêtre divisible par une puissance cinquième > 1. Comme corollaire, nous obtenons que RH est vraie ssi chaque entier naturel divisible par une puissance cinquième > 1 vérifie l'inégalité de Robin.

Integers, 2011

Let

Accepted by The Ramanujan Journal, 2022 (This version is better than the accepted one)

Robin's 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 $n > 5040$ that are not divisible by some prime between $2$ and $1771559$. We prove that the Robin inequality holds when $\frac{\pi^{2}}{6} \times \log\log n' \leq \log\log n$ for some $n>5040$ where $n'$ is the square free kernel of the natural number $n$. The possible smallest counterexample $n > 5040$ of the Robin inequality implies that $q_{m} > e^{31.018189471}$, $1 < \frac{(1 + \frac{1.2762}{\log q_{m}}) \times \log(1.006479799241)}{\log \log n}+ \frac{\log N_{m}}{\log n}$, $(\log n)^{\beta} < 1.000208229291\times\log(N_{m})$ and $n < (1.006479799241)^{m} \times N_{m}$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$, $q_{m}$ is the largest prime divisor of $n$ and $\beta = \prod_{i = 1}^{m} \frac{q_{i}^{a_{i}+1}}{q_{i}^{a_{i}+1}-1}$ when $n$ is an Hardy-Ramanujan integer of the form $\prod_{i=1}^{m} q_{i}^{a_{i}}$.

Zenodo (CERN European Organization for Nuclear Research), 2021

Let P be the set of all primes, ψ(n) = n n∈P,p|n 1 + 1 p be the Dedekind psi function, we unconditionally show that e γ log log n > ψ(n) n for any n > 30, where γ if Euler's constant.

2016

In this paper, we briefly review most of accomplished research in Riemann Zeta function and Riemann hypothesis since Riemann’s age including Riemann hypothesis equivalences as well. We then make use of Robin and Lagarias’ criterions to prove Riemann hypothesis. The goal is, applying Lagarias criterion for n ≥ 1 since Lagarias criterion states that Riemann hypothesis holds if and only if the inequality ∑ d|n d ≤ Hn+exp(Hn) log(Hn) holds for all n ≥ 1. Although, Robin’s criterion is applied as well. Our approach breaks up the set of natural numbers into three main subsets. The first subset is {n ∈ N| 1 ≤ n ≤ 5040}. The second one is {n ∈ N| 5041 ≤ n ≤ 19685} and the third one is {n ∈ N| n ≥ 19686}. In our proof, the third subset for even integers is broken up into odd integer class number sets. Then, mathematical arguments are stated for each odd integer class number set. Odd integer class number set is introduced in this paper. Since the Lagarias Email address: Corresponding author: ...

An extension of Cramér's result gn = O(√ pn log pn) shows a smaller gap between two consecutive primes for large values of n. We prove that if the Riemann hypothesis holds, then there exists N such that n ≥ N implies that pn+1 − pn < 2pn 1/2+(n) where pn is the n-th prime, (n) < 0.0001 and limn→∞ (n) = 0. Our result is a combination of: a) Cramér's work on the distribution of prime numbers, and b) a new theorem stating that, if pn and pn+1 are two consecutive primes and m is their midpoint, then every x in the interval [ log (m−pn) log pn , 1) satisfies pn+1 − pn ≤ 2pn x .

2021

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 1 2. For every prime number p n , we define the sequence X n = q≤p n q q−1 − e γ × log θ(p n), where θ(x) is the Chebyshev function and γ ≈ 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.

Nicolas criterion for the Riemann Hypothesis is based on an inequality that Euler totient function must satisfy at primorial numbers. A natural approach to derive this inequality would be to prove that a specific sequence related to that bound is strictly decreasing. We show that, unfortunately, this latter fact would contradict Cramér conjecture on gaps between consecutive primes. An analogous situation holds when replacing Euler totient by Dedekind Ψ function.

Journal of the Australian Mathematical Society, 2018

Let $h(n)$ denote the largest product of distinct primes whose sum does not exceed $n$. The main result of this paper is that the property for all $n\geq 1$, we have $\log h(n)&lt;\sqrt{\text{li}^{-1}(n)}$ (where $\text{li}^{-1}$ denotes the inverse function of the logarithmic integral) is equivalent to the Riemann hypothesis.

The American Mathematical Monthly, 2002