  The Riemann Hypothesis has been of central interest to mathematicians for a
long time and many unsuccessful attempts have been made to either prove or
disprove it. Since the Riemann zeta function is defined as a sum of the
infinite number of items, in this paper, we look at the Riemann Hypothesis
using a new applied approach to infinity allowing one to easily execute
numerical computations with various infinite and infinitesimal numbers in
accordance with the principle `The part is less than the whole' observed in the
physical world around us. The new approach allows one to work with functions
and derivatives that can assume not only finite but also infinite and
infinitesimal values and this possibility is used to study properties of the
Riemann zeta function and the Dirichlet eta function. A new computational
approach allowing one to evaluate these functions at certain points is
proposed. Numerical examples are given. It is emphasized that different
mathematical languages can be used to describe mathematical objects with
different accuracies. The traditional and the new approaches are compared with
respect to their application to the Riemann zeta function and the Dirichlet eta
function. The accuracy of the obtained results is discussed in detail.
