  We study the homological properties of random simplicial complexes. In
particular, we obtain the asymptotic behavior of lifetime sums for a class of
increasing random simplicial complexes; this result is a higher-dimensional
counterpart of Frieze's $\zeta(3)$-limit theorem for the Erd\H{o}s-R\'{e}nyi
graph process. The main results include solutions to questions posed in an
earlier study by Hiraoka and Shirai about the Linial-Meshulam complex process
and the random clique complex process. One of the key elements of the arguments
is a new upper bound on the Betti numbers of general simplicial complexes in
terms of the number of small eigenvalues of Laplacians on links. This bound can
be regarded as a quantitative version of the cohomology vanishing theorem.
