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Abstract

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