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A probabilistic study on the value-distribution of Dirichlet series attached to certain cusp forms

  • Kohji Matsumoto

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The existence of the asymptotic probability measure of the Riemann zeta-function was proved in Bohr-Jessen’s classical paper [3] [4].

Let s = σ + it be a complex variable, ζ(s) the Riemann zeta-function, and R an arbitrary rectangle with the edges parallel to the axes. Then, for any σ0 > ½ and T > 0, the set

is Jordan measurable, and we denote the Jordan measure of this set by V(T, R; ξ). Then, Bohr-Jessen’s main result asserts the existence of the limit

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References

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[1] Bohr, H., Zur Theorie der Riemann’schen Zetafunktion im kritischen Streifen, Acta Math., 40 (1915), 67100.
[2] Bohr, H. and Courant, R., Neue Anwendungen der Theorie der Diophantischen Approximationen auf die Riemannsche Zetafunktion, J. Reine Angew. Math., 144 (1914), 249274.
[3] Bohr, H. and Jessen, B., Über die Wertverteilung der Riemannschen Zetafunktion, Erste Mitteilung, Acta Math., 54 (1930), 135.
[4] Bohr, H. and Jessen, B., Zweite Mitteilung, ibid., 58 (1932), 155.
[5] Bohr, H. and Jessen, B., Om Sandsynlighedsfordelinger ved Addition af konvekse Kurver, Dan. Vid. Selsk. Skr. Nat. Math. Afd., (8) 12 (1929), 182. = Collected Mathematical Works of H. Bohr, vol. III, 325406.
[6] Borchsenius, V. and Jessen, B., Mean motions and values of the Riemann zeta function, Acta Math., 80 (1948), 97166.
[7] Carlson, F., Contributions à la théorie des séries de Dirichlet, Note I, Arkiv for Mat. Astr. och Fysik 16, no. 18 (1922), 19 pp.
[8] Deligne, P., La conjecture de Weil I, Publ. Math. IHES, 43 (1974), 273307.
[9] Deligne, P. and Serre, J.-P., Formes modulaires de poids 1, Ann. Sci. Ecole Norm. Sup. (4) 7 (1974), 507530.
[10] Good, A., Approximative Funktionalgleiehungen und Mittelwertsätze für Dirichle-treihen, die Spitzformen assoziiert sind, Comment. Math. Helv., 50 (1975), 327361.
[11] Hecke, E., Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung I, Math. Ann., 114 (1937), 128.
[12] Itô, K., Introduction to probability theory, Cambridge Univ. Press 1984.
[13] Jessen, B. and Wintner, A., Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc, 38 (1935), 4888.
[14] Matsumoto, K., Discrepancy estimates for the value-distribution of the Riemann zeta-function III, Acta Arith., 50 (1988), 315337.
[15] Potter, H. S. A., The mean values of certain Dirichlet series I, Proc. London Math. Soc, 46 (1940), 467478.
[16] Prokhorov, Yu. V., Convergence of random processes and limit theorems in probability theory, Teor. Veroyatnost. i Primenen., 1 (1956), 177238. = Theory of Probab. Appl., 1 (1956), 157214.
[17] Rankin, R. A., Contributions to the theory of Ramanujan’s function τ(n) and similar arithmetical functions II, Proc. Cambridge Phil. Soc, 35 (1939), 357372.
[18] Titchmarsh, E. C., The theory of functions, 2nd ed., Oxford 1939.
[19] Titchmarsh, E. C., The theory of the Riemann zeta-function, Oxford 1951.
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A probabilistic study on the value-distribution of Dirichlet series attached to certain cusp forms

  • Kohji Matsumoto

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