A limit theorem for Bohr-Jessen's probability measures of the Riemann zeta-function

Abstract

The asymptotic behavior of value distribution of the Riemann zeta-function ζ(s) is determined for 1 2 < (s) < 1. Namely, the existence is proved, and the value is given, of the limitfor 1 2 < σ < 1, where R( ) is a square in the complex plane C of side length 2 centered at 0, andwhere µ 1 is the one-dimensional Lebesgue measure. Analogous results are obtained also for the Dedekind zeta-functions of Galois number fields. As an essential step, a limit theorem for a sum of independent random vari-r n X n is proved, where X n , n ∈ N, have identical distribution on a finite interval with mean zero, and {r n } is a regularly varying sequence of index −σ. The limit theorem states the convergence ofand gives the explicit value of the limit. In particular, it is shown that the value depends only on σ and is otherwise independent of {r n }.0 1991 Mathematics Subject Classification. 11M06, 60G50