The Riemann Hypothesis

EasyChair Preprint no. 3708, version 5

4 pagesDate: July 29, 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 $\frac{1}{2}$. Many consider it to be the most important unsolved problem in pure mathematics. The Robin's inequality is true for every natural number $n > 5040$ if and only if the Riemann hypothesis is true. We state the following hypothesis: Given two natural numbers $a, b > 5040$, if $3 \times \log \log a > 2 \times \log \log b$, then we obtain that $3 \times \log \log (a + 1) > 2 \times \log \log (b + 2)$. We demonstrate if this hypothesis is true and the Robin's inequality is false for some natural number $n > 5040$, then the Robin's inequality will have an infinite number of counterexamples. However, the Robin's inequality cannot have an infinite number of counterexamples according to the asymptotic growth rate of the sigma function. In this way, we prove if this hypothesis is true, then the Riemann hypothesis is true as well. Consequently, the Riemann hypothesis is true, because of our hypothesis is trivially true.

Keyphrases: Divisor, inequality, number theory