Dirichlet series induced by the Riemann zeta-function

The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on T^ω, the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form ({a_p}, s) = π_p (1 - a _pP^-s)^-1 for {a_p} in T ^ω. Among other things, using the Haar measure on T ^ω for measuring the asymptotic behavior of ζ(s) in the critical strip, we shall prove, in a weak sense, the mean-value theorem for ζ(s), equivalent to the Lindelöf hypothesis.