A+ CATEGORY SCIENTIFIC UNIT

Dirichlet series induced by the Riemann zeta-function

Volume 187 / 2008

Jun-ichi Tanaka Studia Mathematica 187 (2008), 157-184 MSC: Primary 43A17, 46J15, 28D10; Secondary 11M41, 11K70. DOI: 10.4064/sm187-2-4

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.

Authors

  • Jun-ichi TanakaDepartment of Mathematics
    University of North Carolina
    Chapel Hill, NC 27599-3250, U.S.A.
    and
    Department of Mathematics
    School of Education
    Waseda University
    Shinjuku, Tokyo 169-8050, Japan
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image