We consider an arbitrary locally compact abelian group G, with an ordered dual group Γ, acting on a space of measures. Under suitable conditions, we define the notion of analytic measures using the representation of G and the order on Γ. Our goal is to study analytic measures by applying a new transference principle for subspaces of measures, along with results from probability and Littlewood-Paley theory. As a consequence, we derive new properties of analytic measures as well as extensions of previous...

The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on ${}^{\omega }$, the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form $(a_p,s)={\prod }_p{(1-a_p{p}^{-s})}^{-1}$ for $a_p$ in ${}^{\omega }$. Among other things, using the Haar measure on ${}^{\omega }$ 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.