We discuss the computation of regulators and class numbers of hyperelliptic function fields with the help of truncated Euler products. Hereby, we provide tight estimates for the divisor class number of a hyperelliptic function field, i.e. the cardinality of the Jacobian of the corresponding hyperelliptic curve. These estimates can be used to develop an effective method of computing the divisor class number.

Furthermore, we provide experimental data and heuristics on the distribution of the divisor class number within the bounds on the divisor class number. These heuristics are based on recent results by Katz and Sarnak. Our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest.