Adelically summable normalized weights and adelic equidistribution of effective divisors having small diagonals and small heights on the Berkovich projective lines

We introduce the notion of an adelically summable normalized weight $g$, which is a family of normalized weights on the Berkovich projective lines satisfying a summability condition. We then establish an adelic equidistribution of effective $k$-divisors on the projective line over the separable closure $k_s$ in $\overline{k}$ of a product formula field $k$ having small $g$-heights and small diagonals. This equidistribution result generalizes Ye's for the Galois conjugacy classes of algebraic numbers with respect to quasi-adelic measures.