Journal of Mathematical Sciences

, Volume 193, Issue 1, pp 136–144 | Cite as

Extreme Values of Automorphic L-Functions

  • O. M. Fomenko


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,
$$ \log \left| {L\left( {\frac{1}{2}+it,f\times f} \right)} \right|={\varOmega_{+}}\left( {{{{\left( {\frac{{\log t}}{{\log\;\log t}}} \right)}}^{1/2 }}} \right) $$
$$ \log \left| {L\left( {{\sigma_0}+it,f\times f} \right)} \right|={\varOmega_{+}}\left( {{{{\left( {\frac{{\log t}}{{\log\;\log t}}} \right)}}^{{1-{\sigma_0}}}}} \right) $$
For a fixed σ 0\( \left( {\frac{1}{2},1} \right) \). Bibliography: 15 titles.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. C. Titchmarsh, The Theory of the Riemann Zeta Function, 2nd ed., revised by D. R. Heath-Brown, New York (1986).Google Scholar
  2. 2.
    K. Ramachandra, “On the frequency of Titchmarsh’s phenomenon for ς(s),” J. London Math. Soc. (2), 8, 683–690 (1974).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    R. Balasubramanian and K. Ramachandra, “On the frequency of Titchmarsh’s phenomenon ς(s). III,” Proc. Indian Acad. Sci., 86A, 341–351 (1977).MathSciNetGoogle Scholar
  4. 4.
    N. Levinson, “Ω-theorems for the Riemann zeta-function,” Acta Arithm., 20, 317–330 (1972).MathSciNetMATHGoogle Scholar
  5. 5.
    H. L. Montgomery, “Extreme values of the Riemann zeta-function,” Comment. Math. Helv., 52, 511–518 (1977).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    K. Soundararajan, “Extreme values of zeta and L-functions,” Math. Ann., 342, 467–486 (2008).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    O. M. Fomenko, “Fractional moments of automorphic L-functions. II,” Zap. Nauchn. Semin. POMI, 383, 179–192 (2010).MathSciNetGoogle Scholar
  8. 8.
    K. Matsumoto, “Liftings and mean value theorems for automorphic L-functions,” Proc. London Math. Soc. (3), 90, 297–320 (2005).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    O. M. Fomenko, “Fractional moments of automorphic L-functions,” Algebra Analiz, 22, No. 2, 204–224 (2010).MathSciNetGoogle Scholar
  10. 10.
    A. Ivić, The Riemann Zeta-Function, New York (1985).Google Scholar
  11. 11.
    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).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    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.Google Scholar
  13. 13.
    J. W. S. Cassels and A. Frölich, Eds., Algebraic Number Theory, Academic Press (1990).Google Scholar
  14. 14.
    M. Koike, “Higher reciprocity law, modular forms of weight 1, and elliptic curves,” Nagoya Math. J., 98, 109–115 (1985).MathSciNetMATHGoogle Scholar
  15. 15.
    C. J. Moreno, “The Hoheisel phenomenon for generalized Dirichlet series,” Proc. Amer. Math. Soc., 40, 47–51 (1973).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia

Personalised recommendations