Value-distribution of zeta-functions. (English) Zbl 0705.11050

Matsumoto, Kohji

Analytic number theory, Proc. Jap.-Fr. Symp., Tokyo/Jap. 1988, Lect. Notes Math. 1434, 178-187 (1990).

[For the entire collection see Zbl 0698.00019.]
This paper considers certain generalized zeta-functions \(\Phi\) (s) defined for \(Re(s)>1\) by Euler products. It is assumed that \(\Phi\) (s) may be continued into the region Re(s)\(\leq 1\). Imposing some mild restrictions on \(\Phi\) (s), it is shown, for suitable \(\sigma_ 0>\), that for any closed rectangle R with edges parallel to the axes, the function \[ (1/T)mes\{t\in [-T,T]: \log \Phi (\sigma_ 0+it)\in R\} \] tends to a limit as \(T\to \infty\). Results of this type are well-known, but the proof is said to be particularly simple.
One of the side conditions required is that \(\int^{T}_{-T}| \Phi (\sigma_ 0+it)|^ 2 dt\ll T\). Unfortunately this means that one cannot handle the case in which \(\Phi\) (s) is a general Dedekind zeta- function without imposing further restrictions on \(\sigma_ 0\).

Reviewer: D.R.Heath-Brown

MSC:

11M41

Other Dirichlet series and zeta functions

Keywords:

value-distribution; existence of asymptotic probability measure; generalized zeta-functions; Euler products

Citations:

Zbl 0698.00019