  We present a remarkably simple and surprisingly natural interpretation of the
values of zeta functions at negative integers and zero. Namely we are able to
relate these values to areas related to partial sums of powers. We apply these
results to further interpretations of values of $L$-functions at negative
integers. We hint in a very brief way at some expected connections of this work
with other current efforts to understand the mysteries of the values of zeta
functions at integers.
