# zbMATH — the first resource for mathematics

On the magnitude of asymptotic probability measures of Dedekind zeta- functions and other Euler products. (English) Zbl 0715.11043
Let $$\phi(s)$$ be a certain zeta-function, $$R$$ an arbitrary closed rectangle in the complex plane with the edges parallel to the axes, and $$V(T,R,\sigma;\phi)$$ the length of the set $\{t\in [-T,T] | \quad \log \phi (\sigma +it)\in {\mathbb{R}}\}$ for some fixed $$\sigma$$. The existence of the limit $W(R,\sigma;\phi)=\lim_{T\to \infty}V(T,R,\sigma;\phi)/2T$ for fairly general zeta-functions has been proved in the author’s paper “Value-distribution of zeta- functions” [Lect. Notes Math. 1434, 178-187 (1990; Zbl 0705.11050)].
We restrict ourselves to the case $$R=R(r)=\{z |$$-r$$\leq Re(z)\leq r$$, $$-r\leq Im(z)\leq r\}$$, and consider the quantity $$W(r,\sigma;\phi)=1- W(R(r),\sigma;\phi)$$. In the case of the Riemann zeta-function $$\zeta(s)$$, D. Joyner [Distribution theorems of L-functions (Pitman Research Notes Math. Ser. 142) (1986; Zbl 0609.10032)] has determined the real order of $$W(r,\sigma;\zeta)$$ up to constant factors.
In the present article, we extend Joyner’s idea to a general principle based on Montgomery’s inequality on the sums of independent random variables. As a corollary, we determine the order, up to constant factors, of $$W(r,\sigma;\zeta_ F)$$ for Dedekind zeta-functions $$\zeta_ F(s)$$ attached to an arbitrary algebraic number field $$F$$. The case of the zeta-functions associated with certain cusp forms is also discussed.
Reviewer: K.Matsumoto

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11R42 Zeta functions and $$L$$-functions of number fields 11K99 Probabilistic theory: distribution modulo $$1$$; metric theory of algorithms 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations
##### Citations:
Zbl 0705.11050; Zbl 0609.10032
Full Text: