Dirichlet series induced by the Riemann zeta-function

• Autorzy: Jun-ichi Tanaka
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11K70: Harmonic analysis and almost periodicity
• 28D10: One-parameter continuous families of measure-preserving transformations
• 11M41: Other Dirichlet series and zeta functions
• 43A17: Analysis on ordered groups, H p -theory
• 46J15: Banach algebras of differentiable or analytic functions, H p -spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2008
• Tom: 187
• Numer: 2
• Strony: 157-184

Daty

wydano

2008

Twórcy

autor

Jun-ichi Tanaka

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

Identyfikatory

DOI

10.4064/sm187-2-4