Dirichlet series induced by the Riemann zeta-function

Junichi Tanaka*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    1 Citation (Scopus)


    The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on Tω, the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form ({ap}, s) = πp (1 - a pP-s)-1 for {ap} 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.

    Original languageEnglish
    Pages (from-to)157-184
    Number of pages28
    JournalStudia Mathematica
    Issue number2
    Publication statusPublished - 2008


    • Dirichlet series
    • Lindelöf hypothesis
    • Mean-value theorems
    • Outer functions
    • Riemann zeta-function

    ASJC Scopus subject areas

    • Mathematics(all)


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