  We develop a general description of the superconductivity of lattice fermions
based on the BCS theory. We propose a modeling of the density of states (DOS)
of lattice fermions, where divergent and semi-metallic structures are described
by asymptotic expansions around the Fermi energy. This modeling leads to a
unified representation of the transition temperature $T_c$ at half filling,
which reproduces asymptotic forms of $T_c$ derived in several lattices, such as
the square, honeycomb, and Lieb lattices, for the weakly interacting limit. The
derived asymptotic forms of $T_c$ are categorized into four types, which is
attributed to the different responses of the degenerate fermions depending on
the DOS structures. The DOS with a delta-functional singularity induces the
highest $T_c$ in the weakly interacting region, where $T_c$ is linearly
proportional to the pairing interaction $U$. Three kinds of universal ratios
defined in the BCS theory no longer reduce to constants independent of the
system parameters but can be parameterized with a certain variable that
characterizes the singular structures of the DOS. We find universal
relationship among thermodynamic quantities that holds for all parameter
regions. Further, we numerically demonstrate that in multi-band systems the
correlation effects can induce an effective delta-functional singularity. This
phenomenon generally appears in multi-energy (or multi-gap) systems and may
provide a plausible guideline for material designs of high-$T_c$
superconductor.
