This paper is a summary of results on the PG(s) function, which is the reciprocal of the probabilistic zeta function for finite groups. This function gives the probability that s randomly chosen elements generate a group G, and information about the structure of the group G is embedded in it.
Benesh B. 2010. The probabilistic zeta function. In: Kappe L-C, Magidin A, Morse RF, editors. Contemporary Mathematics 511: Computational Group Theory and the Theory of Groups, II, Harlaxton Conference on Computational Group Theory and Cohomology, August 4-8, 2008, Harlaxton College, Grantham, United Kingdom. Providence (RI): American Mathematical Society. p. 1-9.