On zeta-functions associated to certain cusp forms. I

A. Laurinčikas 1  and J. Steuding 2
  • 1 Vilnius University
  • 2 Johann Wolfgang Goethe-Universität Frankfurt

Abstract

In the paper the asymptotics for Dirichlet polynomials associated to certain cusp forms are obtained.

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  • [1] H. Bohr: “Über Diophantische Approximationen und ihre Anwendung auf Dirichletsche Reihen, besonders auf die Riemannsche Zetafunktion”, In: Proc. 5th Congress of Scand. Math., 1923, Helsingfors, pp. 131–154.

  • [2] H. Bohr and B. Jessen: “Über die Wertverteilung der Riemannschen Zetafunktion, Erste Mitteilung”, Acta Math., Vol. 54, (1930), pp. 1–35.

  • [3] H. Bohr and B. Jessen: “Über die Wertverteilung der Riemannschen Zetafunktion, Zweite Mitteilung”, Acta Math., Vol. 58, (1932), pp. 1–55.

  • [4] P. Deligne: “La conjecture de Weil”, Inst. Hautes Etudes Sci. Publ. Math., Vol. 43, (1974), pp. 273–307.

  • [5] F. Grupp: “Eine Bemerkung zur Ramanujanschen τ-Funktion”, Arch. Math., Vol. 43, (1984), pp. 358–363. http://dx.doi.org/10.1007/BF01196660

  • [6] D.R. Health-Brown: “Franctional moments of the Riemann zeta-function”, J. London Math. Soc., Vol. 24(2), (1981), pp. 65–78.

  • [7] D. Joyner: Distribution Theorems of L-functions, Longman Scientific, Harlow, 1986.

  • [8] A. Kačėnas and A. Laurinčikas: “On Dirichlet series related to certain cusp forms”, Lith. Math. J., Vol. 38, (1998), pp. 64–76. http://dx.doi.org/10.1007/BF02465545

  • [9] A. Laurinčikas: Limit Theorems for the Riemann Zeta-function, Kluwer, Dordrecht, 1996.

  • [10] A. Laurinčikas and R. Garunkštis: The Lerch Zeta-Function, Kluwer, Dordrecht, 2002.

  • [11] R. Leipnik: “The lognormal distribution and strong nonuniqueness of the moment problems”, Teor. Veroyatn. Primenen., Vol. 26, (1981), pp. 863–865.

  • [12] B.V. Levin and A.S. Fainleib: “On one method of summing of multiplicative functions”, Izv. AN SSSR, ser. matem., Vol. 31, (1967), pp. 697–710.

  • [13] B.V. Levin and A.S. Fainleib: “Application of certain integral equations to problems of the number theory”, Uspechi matem. nauk, Vol. 22(3), (1967), pp. 119–197.

  • [14] K. Matsumoto: “Probabilistic value-distribution theory of zeta-functions”, Sūgaku, Vol. 53, (2001), pp. 279–296.

  • [15] H.L. Montgomery and R.C. Vaughan: “Hilbert’s inequality”, J. London Math. Soc., Vol. 8(2), (1974), pp. 73–82.

  • [16] L. Mordell: “On Mr. Ramanujan’s empirical expansions of modular functions”, Proc. Camb. Phil. Soc., Vol. 19, (1917), pp. 117–124.

  • [17] S. Ramanujan: “On certain arithmetic al functions”, it Trans. Camb. Phil. Soc., Vol. 22, (1916), pp. 159–184.

  • [18] R.A. Rankin: “An Ω-result for the coefficients of cusp forms”, Math. Ann., Vol. 203, (1973), pp. 239–250. http://dx.doi.org/10.1007/BF01629259

  • [19] R.A. Rankin: “Ramanujan’s tau-function and its generalizations”, Ramanujan revisited (Urbana-Champaign, Ill. 1987), Academic Press, Boston, 1988, pp. 245–268.

  • [20] A. Selberg: “Old and new conjectures and results about a class of Dirichlet series”, Collected Papers, Vol. 2, Springer-Verlag, Berlin, 1991, pp. 47–63.

  • [21] G. Tenenbaum: Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, 1995.

  • [22] E. Wirsing: “Das asymptotische Verhalten von Summen Über multiplikative Funktionen”, Math. Ann., Vol. 143, (1961), pp. 75–102. http://dx.doi.org/10.1007/BF01351892

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