Abstract
We prove upper and lower bounds for fractional moments of zeta-functions attached to certain cusp forms on the critical line; the upper bound being conditional subject to the truth of the Riemann hypothesis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Deligne, P.: La Conjecture de Weil I, II. Publ. Math. IHES 43, 273–307 (1974); 52, 137–252 (1980)
Gabriel, R.M.: Some results concerning the integrals of moduli of regular functions along certain curves. J. Lond. Math. Soc. 2, 112–117 (1927)
Good, A.: Ein Mittelwertsatz für Dirichletreihen, die zu Modulformen assoziiert sind. Comment. Math. Helv. 49, 35–47 (1974)
Good, A.: Approximative Funktionalgleichungen und Mittelwertsätze für Dirichletreihen, die zu Spitzenformen assoziiert sind. Comment. Math. Helv. 50, 327–361 (1975)
Good, A.: Beiträge zur Theorie der Dirichletreihen, die Spitzenformen zugeordnet sind. J. Number Theory 13, 18–65 (1981)
Good, A.: The mean square of Dirichlet series associated with cusp forms. Mathematika 29, 278–295 (1982)
Heath-Brown, D.R.: Fractional moments of the Riemann zeta-function. J. Lond. Math. Soc. (2) 24, 65–78 (1981)
Jutila, M.: Lectures on a Method in the Theory of Exponential Sums. Lectures on Math. and Phys., vol. 80. Tata Inst. Fund. Res., Bombay (1987)
Laurinčikas, A.: Limit Theorems for the Riemann Zeta-Function. Kluwer, Dordrecht (1996)
Laurinčikas, A., Steuding, J.: On zeta-functions associated to certain cusp forms. I. Central Eur. J. Math. 2, 1–18 (2004)
Laurinčikas, A., Zamarys, S.: Asymptotics for the mean value of normalized zeta-functions of certain cusp forms. Integral Transform. Special Funct. 16(5–6), 437–450 (2005)
Matsumoto, K.: A probabilistic study on the value-distribution of Dirichlet series attached to certain cusp forms. Nagoya Math. J. 116, 123–138 (1989)
Matsumoto, K.: Recent developments in the mean square theory of the Riemann zeta and other zeta-functions. In: Number Theory, Trends Math., pp. 241–286. Birkhäuser, Boston (2000)
Montgomery, H.L., Vaughan, R.C.: Hilbert’s inequality. J. Lond. Math. Soc. 8(2), 73–82 (1974)
Potter, H.S.A.: The mean values of certain Dirichlet series. I. Proc. Lond. Math. Soc. (2) 46, 467–478 (1940)
Potter, H.S.A.: The mean values of certain Dirichlet series. II. Proc. Lond. Math. Soc. (2) 47, 1–19 (1940)
Rankin, R.A.: Contributions to the theory of Ramanujan’s function τ(n), and similar arithmetical functions. II: The order of the Fourier coefficients of integral modular forms. Proc. Camb. Philos. Soc. 35, 357–372 (1939)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Laurinčikas, A., Steuding, J. On Fractional Power Moments of Zeta-Functions Associated with Certain Cusp Forms. Acta Appl Math 97, 25–39 (2007). https://doi.org/10.1007/s10440-007-9138-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-007-9138-6