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Asymptotic probability measures of zeta-functions of algebraic number fields. (English) Zbl 0746.11051
Let \(K\) be an algebraic number field of degree \(l\) and \(\zeta_ K(s)\) its Dedekind zeta-function. The author considers the value distribution of \(\log \zeta_ K(s)\) on vertical lines in the half-plane \(\sigma > 1- L^{-1}\), where \(L=\max(l,2)\). It is shown that given a rectangle \(R\) in the complex plane and letting \(s\) run over a vertical line, \(\log \zeta_ K(s)\) lies in \(R\) with a certain asymptotic probability \(W(R)\). Moreover, the rate of convergence to this asymptotic probability is estimated.

11R42 Zeta functions and \(L\)-functions of number fields
11M41 Other Dirichlet series and zeta functions
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
Full Text: DOI
[1] Bohr, H, Zur theorie der Riemann’schen zetafunktion im kritischen streifen, Acta math., 40, 67-100, (1915) · JFM 45.0719.01
[2] Bohr, H; Courant, R, Neue anwendungen der theorie der diophantischen approximationen auf die riemannsche zetafunktion, J. reine angew. math., 144, 249-274, (1914) · JFM 45.0718.02
[3] Bohr, H; Jessen, B, Über die wertverteilung der riemannschen zetafunktion, erste mitteilung, Acta math., 54, 1-35, (1930) · JFM 56.0287.01
[4] Bohr, H; Jessen, B, Über die wertverteilung der riemannschen zetafunktion, zweite mitteilung, Acta math., 58, 1-55, (1932) · JFM 58.0321.02
[5] Bohr, H; Jessen, B, Om sandsynlighedsfordelinger ved addition af konvekse kurver, Dan. vid. selsk. skr. nat. math. afd., 12, 8, 1-82, (1929), [“Collected Mathematical Works of H. Bohr,” Vol. III, pp. 325-406] · JFM 55.1068.02
[6] Borchsenius, V; Jessen, B, Mean motions and values of the Riemann zeta function, Acta math., 80, 97-166, (1948) · Zbl 0038.23201
[7] Carlson, F, Contributions à la théorie des séries de Dirichlet, note I, Ark. mat. astr. fysik, 16, No. 18, 19, (1922) · JFM 48.0338.02
[8] Chandrasekharan, K; Narasimhan, R, The approximate functional equation for a class of zeta-functions, Math. ann., 152, 30-64, (1963) · Zbl 0116.27001
[9] Itô, K, ()
[10] Jessen, B; Wintner, A, Distribution functions and the Riemann zeta function, Trans. amer. math. soc., 38, 48-88, (1935) · JFM 61.0462.03
[11] Matsumoto, K, Discrepancy estimates for the value-distribution of the Riemann zeta-function I, Acta arith., 48, 167-190, (1987) · Zbl 0569.10019
[12] Matsumoto, K, Discrepancy estimates for the value-distribution of the Riemann zeta-function. III, Acta arith., 50, 315-337, (1988) · Zbl 0657.10042
[13] Matsumoto, K, A probabilistic study on the value-distribution of Dirichlet series attached to certain cusp forms, Nagoya math. J., 116, 123-138, (1989) · Zbl 0675.10017
[14] Matsumoto, K, Value-distribution of zeta-functions, (), 178-187
[15] Matsumoto, K; Miyazaki, T, On some hypersurfaces of high-dimensional tori related with the Riemann zeta-function, Tokyo J. math., 10, 271-279, (1987) · Zbl 0638.10041
[16] Potter, H.S.A, The Mean values of certain Dirichlet series I, (), 467-478 · Zbl 0025.26301
[17] Titchmarsh, E.C, ()
[18] Waldschmidt, M, Une mesure de transcendance de en, Sém. delange-Pisot-poitou, 17, G4, 5, (1975/1976)
[19] Waldschmidt, M, Transcendence measures for exponentials and logarithms, J. austral. math. soc. ser. A, 25, 445-465, (1978) · Zbl 0388.10022
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