Robin's inequality,Lagarias' criterion, and Riemann hypothesis

Abstract

In this paper, we 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 the two main subsets. The first subset is $\{n\in \mathbb{N}| ~ 1\leq n\leq 2(3\times5\dots\times331)^{2}\}$. The second one is $\{n\in \mathbb{N}| ~ n\geq 2(3\times5\dots\times331)^{2}\}$. In our proof, the second subset is decomposed again into the three sub-subsets including odd numbers and the two groups of the even numbers. Then,each group of the even numbers is expressed by an odd integer class number set. Finally, 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 and Thomas Morrill's paper, we do prove it for the second subset using both Lagarias and Robin's criteria and mathematical arguments. It then follows that Riemann hypothesis holds as well. Essential keys of the proof for large numbers are theorem 1 proving $\sigma(m)<\frac{1}{2}e^{\gamma}m \log\log(2m)$ for odd numbers $m\geq (3\times5\dots\times331)^{2}$, lemma9 and lemma 10 proving $e^{\gamma}(1-\frac{1}{p_{1}})\dots (1-\frac{1}{p_{n}})\log\log(2p_{1}\dots p_{n})<2$ for $n\geq 1$ and $e^{\gamma}(1-\frac{1}{p_{1}})\dots (1-\frac{1}{p_{n}})\log\log(2p^{2}_{1}\dots p^{2}_{n})>2$ for $n\geq 66$.