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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
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