A Computational Evidence of the Riemann Hypothesis

EasyChair Preprint no. 3708, version 33

12 pagesDate: November 30, 2020

Abstract

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 $\frac{1}{2}$. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all sufficiently large $n$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. In 1984, Guy Robin proved that the inequality is true for all $n > 5040$ if and only if the Riemann Hypothesis is true. In 2002, Lagarias proved that if the inequality $\sigma(n) \leq H_{n} + exp(H_{n}) \times \log H_{n}$ holds for all $n \geq 1$, then the Riemann Hypothesis is true, where $H_{n}$ is the $n^{th}$ harmonic number. We show certain properties of these both inequalities that leave us to a verified proof of the Riemann Hypothesis. These results are supported by the claim that a numerical computer calculation verifies that the subtraction of
$\log (e^{\gamma} \times q_{m} \times r) + e^{\gamma} \times q_{m} \times r \times \log \log (e^{\gamma} \times q_{m} \times r)$
with
$(q_{m} + 1) \times \log (e^{\gamma} \times (r + 1)) + (q_{m} + 1) \times e^{\gamma} \times (r + 1) \times \log \log (e^{\gamma} \times (r + 1))$
is monotonically increasing as much as $q_{m}$ and $r$ become larger just starting with the initial values of $q_{m} = 47$ and $r = 1$, where $q_{m}$ is a prime number and $r$ is a natural number. In this way, we can confirm that the Riemann Hypothesis is true based on computational mathematics using a simple and naive computer assisted proof.

Keyphrases: Divisor, inequality, number theory