Dirichlet series induced by the Riemann zeta-function

Jun-ichi Tanaka. "Dirichlet series induced by the Riemann zeta-function." Studia Mathematica 187.2 (2008): 157-184. <http://eudml.org/doc/284972>.

@article{Jun2008,
abstract = {The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on $^\{ω\}$, the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form $(\{a_\{p\}\},s) = ∏_\{p\} (1-a_\{p\}p^\{-s\})^\{-1\}$ for $\{a_\{p\}\}$ in $^\{ω\}$. Among other things, using the Haar measure on $^\{ω\}$ 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.},
author = {Jun-ichi Tanaka},
journal = {Studia Mathematica},
keywords = {outer functions; Riemann zeta-function; Dirichlet series; mean-value theorems; Lindelöf hypothesis},
language = {eng},
number = {2},
pages = {157-184},
title = {Dirichlet series induced by the Riemann zeta-function},
url = {http://eudml.org/doc/284972},
volume = {187},
year = {2008},
}

TY - JOUR
AU - Jun-ichi Tanaka
TI - Dirichlet series induced by the Riemann zeta-function
JO - Studia Mathematica
PY - 2008
VL - 187
IS - 2
SP - 157
EP - 184
AB - The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on $^{ω}$, the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form $({a_{p}},s) = ∏_{p} (1-a_{p}p^{-s})^{-1}$ for ${a_{p}}$ in $^{ω}$. Among other things, using the Haar measure on $^{ω}$ 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.
LA - eng
KW - outer functions; Riemann zeta-function; Dirichlet series; mean-value theorems; Lindelöf hypothesis
UR - http://eudml.org/doc/284972
ER -