Download PDFOpen PDF in browserThe Riemann Hypothesis Is TrueEasyChair Preprint 7094, version 28 pages•Date: December 1, 2021AbstractThe 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 sumofdivisors function and $\gamma \approx 0.57721$ is the EulerMascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q1} > e^{\gamma} \times \log\theta(q_{n})$ 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. Keyphrases: Chebyshev function, Nicolas inequality, Riemann hypothesis, Riemann zeta function, Robin inequality, prime numbers, sumofdivisors function
