Arguments in Favor of the Riemann Hypothesis

EasyChair Preprint 7134

Abstract

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.

Keyphrases: Chebyshev function, Nicolas inequality, Riemann hypothesis, Riemann zeta function, Robin inequality, prime numbers, sum-of-divisors function

@booklet{EasyChair:7134,
  author    = {Frank Vega},
  title     = {Arguments in Favor of the Riemann Hypothesis},
  howpublished = {EasyChair Preprint 7134},
  year      = {EasyChair, 2021}}