  We study Euler systems for $\mathbb{G}_m$ over a number field $k$. Motivated
by a distribution-theoretic idea of Coleman, we formulate a conjecture
regarding the existence of such systems that is elementary to state and yet
strictly finer than Kato's equivariant Tamagawa number conjecture for Dirichlet
$L$-series at $s=0$. To investigate the conjecture, we develop an abstract
theory of `Euler limits' and, in particular, prove the existence of canonical
`restriction' and `localisation' sequences in this theory. By using this
approach we obtain a variety of new results, ranging from a proof, modulo
standard $\mu$-vanishing hypotheses, of our central conjecture in the case $k$
is $\mathbb{Q}$ or imaginary quadratic to a proof of the `minus part' of Kato's
conjecture in the case $k$ is totally real. In proving these results, we also
show that higher-rank Euler systems for a wide class of $p$-adic
representations control the structure of Iwasawa-theoretic Selmer groups in the
manner predicted by `main conjectures'.
