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Dirichlet series induced by the Riemann zeta-function

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Tanaka J.-i.

Studia Mathematica

2008

tom 187

nr 2

157-184

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 $𝔷({a_{p}},s) = ∏_{p} (1-a_{p}p^{-s})^{-1}$ 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.

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1 Studia Mathematica

1 Tanaka J.-i.

1 2008

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Dirichlet series induced by the Riemann zeta-function