Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers

Robin’s theorem is one of the ingenious reformulation of the Riemann hypothesis (RH). It states that the RH is true if and only if $\sigma (n)<e^{\gamma }n\log \log n$ for all $n>5040$ where $\sigma (n)$ is the sum of divisors of $n$ and $\gamma$ is Euler’s constant. In this paper we show that how the RH is delicate in terms of certain subsets of superabundant numbers, namely extremely abundant numbers and some of its specific supersets.