The Riemann Hypothesis Is Most Likely True

EasyChair Preprint 7007, version 4

Abstract

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.

Keyphrases: Chebyshev function, Nicolas inequality, Riemann hypothesis, Robin inequality, prime numbers

@booklet{EasyChair:7007,
  author    = {Frank Vega},
  title     = {The Riemann Hypothesis Is Most Likely True},
  howpublished = {EasyChair Preprint 7007},
  year      = {EasyChair, 2021}}