  We study the number of lattice points in integer dilates of the rational
polytope $P = (x_1,...,x_n) \in \R_{\geq 0}^n : \sum_{k=1}^n x_k a_k \leq 1$,
where $a_1,...,a_n$ are positive integers. This polytope is closely related to
the linear Diophantine problem of Frobenius: given relatively prime positive
integers $a_1,...,a_n$, find the largest value of t (the Frobenius number) such
that $m_1 a_1 + ... + m_n a_n = t$ has no solution in positive integers
$m_1,...,m_n$. This is equivalent to the problem of finding the largest dilate
tP such that the facet $\sum_{k=1}^n x_k a_k = t$ contains no lattice point. We
present two methods for computing the Ehrhart quasipolynomials of P which count
the integer points in the dilated polytope and its interior. Within the
computations a Dedekind-like finite Fourier sum appears. We obtain a
reciprocity law for these sums, generalizing a theorem of Gessel. As a
corollary of our formulas, we rederive the reciprocity law for Zagier's
higher-dimensional Dedekind sums. Finally, we find bounds for the
Fourier-Dedekind sums and use them to give new bounds for the Frobenius number.
