Dirichlet series induced by the Riemann zeta-function

Jun-ichi Tanaka

Jun-ichi Tanaka

Studia Mathematica, Tome 187 (2008), p. 157-184 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $(a_{p}, s) = \prod_{p} {(1 - a_{p} p^{- s})}^{- 1}$ 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.

Publié le : 2008-01-01

Zbl 1209.43001

EUDML-ID : urn:eudml:doc:284972

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     title = {Dirichlet series induced by the Riemann zeta-function},
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     volume = {187},
     year = {2008},
     pages = {157-184},
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     language = {en},
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Jun-ichi Tanaka. Dirichlet series induced by the Riemann zeta-function. Studia Mathematica, Tome 187 (2008) pp. 157-184. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm187-2-4/