  In this paper we prove a concentration theorem for arithmetic $K_0$-theory,
this theorem can be viewed as an analog of R. Thomason's result in the
arithmetic case. We will use this arithmetic concentration theorem to prove a
relative fixed point formula of Lefschetz type in the context of Arakelov
geometry. Such a formula was conjectured of a slightly stronger form by K.
K\"{o}hler and D. Roessler and they also gave a correct route of its proof.
Nevertheless our new proof is much simpler since it looks more natural and it
doesn't involve too many complicated computations.
