  Srinivasa Ramanujan was a great self-taught Indian mathematician, who died a
century ago, at the age of only 32, one year after returning from England.
Among his numerous achievements is the assignment of sensible, finite values to
divergent series, which correspond to Riemann's $\zeta$-function with negative
integer arguments. He hardly left any explanation about it, but following the
few hints that he gave, we construct a direct justification for the best known
example, based on analytic continuation. As a physical application of Ramanujan
summation we discuss the Casimir effect, where this way of removing a divergent
term corresponds to the renormalization of the vacuum energy density, in
particular of the photon field. This leads to the prediction of the Casimir
force between conducting plates, which has now been accurately confirmed by
experiments. Finally we review the discussion about the meaning and
interpretation of the Casimir effect. This takes us to the mystery surrounding
the magnitude of Dark Energy.
