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

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 limit lim →∞ ( (log )σ)−1/(1−σ) logW (C \R( ), σ, ζ) for 1 2 < σ < 1, where R( ) is a square in the complex plane C of side length 2 centered at 0, and W (A,σ, ζ) = lim T→∞ (2T )−1μ1({t ∈ [−T, T ] | log ζ(σ + t √−1) ∈ A}) , A ⊂ C , where μ1 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 variables X = ∞ ∑ n=1 rnXn is proved, where Xn, n ∈ N, have identical distribution on a finite interval with mean zero, and {rn} is a regularly varying sequence of index −σ. The limit theorem states the convergence of lim N→∞ N−1 log Prob[ X > ∑ n≤N rn ] and gives the explicit value of the limit. In particular, it is shown that the value depends only on σ and is otherwise independent of {rn}.