Dirichlet series induced by the Riemann zeta-function

Jun-ichi Tanaka¹

¹ Department 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

Studia Mathematica, Tome 187 (2008) no. 2, pp. 157-184

Résumé

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.

Détail

DOI : 10.4064/sm187-2-4

Keywords: riemann zeta function zeta extends outer function ergodic hardy spaces mathbb omega infinite dimensional torus indexed primes enables investigate collectively certain properties dirichlet series form mathfrak prod a s mathbb omega among other things using haar measure mathbb omega measuring asymptotic behavior zeta critical strip shall prove weak sense mean value theorem zeta equivalent lindel hypothesis