Abstract
Ω-theorems for some automorphic L-functions and, in particular, for the Rankin−Selberg L-function L(s, f × f) are considered. For example, as t tends to infinity,
and
For a fixed σ 0 ∈ \( \left( {\frac{1}{2},1} \right) \). Bibliography: 15 titles.
Similar content being viewed by others
References
E. C. Titchmarsh, The Theory of the Riemann Zeta Function, 2nd ed., revised by D. R. Heath-Brown, New York (1986).
K. Ramachandra, “On the frequency of Titchmarsh’s phenomenon for ς(s),” J. London Math. Soc. (2), 8, 683–690 (1974).
R. Balasubramanian and K. Ramachandra, “On the frequency of Titchmarsh’s phenomenon ς(s). III,” Proc. Indian Acad. Sci., 86A, 341–351 (1977).
N. Levinson, “Ω-theorems for the Riemann zeta-function,” Acta Arithm., 20, 317–330 (1972).
H. L. Montgomery, “Extreme values of the Riemann zeta-function,” Comment. Math. Helv., 52, 511–518 (1977).
K. Soundararajan, “Extreme values of zeta and L-functions,” Math. Ann., 342, 467–486 (2008).
O. M. Fomenko, “Fractional moments of automorphic L-functions. II,” Zap. Nauchn. Semin. POMI, 383, 179–192 (2010).
K. Matsumoto, “Liftings and mean value theorems for automorphic L-functions,” Proc. London Math. Soc. (3), 90, 297–320 (2005).
O. M. Fomenko, “Fractional moments of automorphic L-functions,” Algebra Analiz, 22, No. 2, 204–224 (2010).
A. Ivić, The Riemann Zeta-Function, New York (1985).
A. Sankaranarayanan and J. Sengupta, “Omega theorems for a class of L-functions (A note on the Rankin−Selberg zeta-function),” Funct. Approx. Comment. Math., 36, 119–131 (2006).
A. Ivić, “On zeta-functions associated with Fourier coefficients of cusp forms,” in: Proceeding of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Salerno (1992), pp. 231–246.
J. W. S. Cassels and A. Frölich, Eds., Algebraic Number Theory, Academic Press (1990).
M. Koike, “Higher reciprocity law, modular forms of weight 1, and elliptic curves,” Nagoya Math. J., 98, 109–115 (1985).
C. J. Moreno, “The Hoheisel phenomenon for generalized Dirichlet series,” Proc. Amer. Math. Soc., 40, 47–51 (1973).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 404, 2012, pp. 233–247.
Rights and permissions
About this article
Cite this article
Fomenko, O.M. Extreme Values of Automorphic L-Functions. J Math Sci 193, 136–144 (2013). https://doi.org/10.1007/s10958-013-1442-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-013-1442-2