  When $X$ is a $d$-dimensional variety defined over a field $k$ of
characteristic zero, a constructive resolution of singularities can be achieved
by successively lowering the maximum multiplicity via blow ups at smooth
equimultiple centers. This is done by stratifying the maximum multiplicity
locus of $X$ by means of the so called {\em resolution functions}. The most
important of these functions is what we know as {\em Hironaka's order function
in dimension $d$}. Actually, this function can be defined for varieties when
the base field is perfect; however if the characteristic of $k$ is positive,
the function is, in general, too coarse and does not provide enough information
so as to define a resolution. It is very natural to ask what the meaning of
this function is in this case, and to try to find refinements that could lead,
ultimately, to a resolution. In this paper we show that Hironaka's order
function in dimension $d$ can be read in terms of the {\em Nash multiplicity
sequences} introduced by Lejeune-Jalabert. Therefore, the function is intrinsic
to the variety and has a geometrical meaning in terms of its space of arcs.
