  We calculate the perturbative corrections to fermion bilinears that are used
in numerical simulations when extracting weak matrix elements using staggered
fermions. This extends previous calculations of Golterman and Smit, and Daniel
and Sheard. In particular, we calculate the corrections for non-local bilinears
defined in Landau gauge with gauge links excluded. We do this for the simplest
operators, i.e. those defined on a $2^4$ hypercube, and for tree level improved
operators which live on $4^4$ hypercubes. We also consider gauge invariant
operators in which the ``tadpole'' contributions are suppressed by projecting
the sums of products of gauge links back in to the gauge group. In all cases,
we find that the variation in the size of the perturbative corrections is
smaller than those with the gauge invariant unimproved operators. This is most
strikingly true for the smeared operators. We investigate the efficacy of the
mean-field method of Lepage and Mackenzie at summing up tadpole contributions.
In a companion paper we apply these results to four-fermion operators.
