Riemann hypothesis equivalences, Robin inequality, Lagarias criterion, and Riemann hypothesis

Abstract: In this paper, we briefly review most of accomplished research in Riemann Zeta function and Riemann hypothesis since Riemann's age including Riemann hypothesis equivalences as well. We then make use of Robin and Lagarias' criteria to prove Riemann hypothesis. The goal is, using Lagarias criterion for $n\geq 1$ since Lagarias criterion states that Riemann hypothesis holds if and only if the inequality $\sum_{d|n}d\leq H_{n}+\exp(H_{n})\log(H_{n})$ holds for all $n\geq 1$. Although, Robin's criterion is used as well. Our approach breaks up the set of the natural numbers into three main subsets. The first subset is ${n\in \mathbb{N}| ~ 1\leq n\leq 5040}$. The second one is ${n\in \mathbb{N}| ~ 5041\leq n\leq 19685}$ and the third one is ${n\in \mathbb{N}| ~ n\geq 19686}$. In our proof, the third subset for even integers is broken up into odd integer class number sets. Then, mathematical arguments are stated for each odd integer class number set. Odd integer class number set is introduced in this paper. Since the Lagarias criterion holds for the first subset regarding computer aided computations, we do prove it using both Lagarias and Robin's criteria for the second and third subsets and mathematical arguments accompanied by a large volume of computer language programs. It then follows that Riemann hypothesis holds as well.