  Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem
states that the Riemann hypothesis is equivalent to the inequality
\sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It
is a natural question in this direction to find a first integer, if exists,
which violates this inequality. Following this process, we introduce a new
sequence of numbers and call it as extremely abundant numbers. In this paper we
show that the Riemann hypothesis is true, if and only if, there are infinitely
many of these numbers. Moreover, we investigate some of their properties
together with superabundant and colossally abundant numbers.

Number Theory