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

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

Zbl 0698.00019