Probability density functions attached to zeta functions

The study of the value-distribution of the Riemann zeta function is a classical topic in analytic number theory. In 1930s, Bohr and Jessen proved the existence of a certain limit value regarded as the probability that values of the Riemann zeta function belong to a given region in the complex plane. After Bohr and Jessen, similar results were proved for many other zeta functions. In this talk, I'll talk about density functions of such probabilities attached to the value-distributions of zeta functions. The density functions, which were named ``M-functions'' by Ihara, are connected with mean values of zeta functions, distributions of zeros of zeta functions, and so on.