Skip to main content
SpringerLink
Log in

A new zero-density result of L-functions attached to Maass forms

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

Let f denote a normalized Maass cusp form for SL(2, ℤ), which is an eigenfunction of all the Hecke operators T(n) as well as the reflection operator \( T_{ - 1} :z \to - \bar z \). We obtain a zero-density result of the L-function attached to f near σ = 1. This improves substantially the previous results in this direction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Huxley, M. N.: On the difference between consecutive primes. Invent. Math., 15, 164–170 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bombieri, E.: On the large sieve. Mathematika, 12, 201–225 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  3. Zhang, D.: Zero-density estimates for automorphic L-functions. Acta Mathematica Sinica, English Series, 25(6), 945–960 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Sankaranarayanan, A., Sengupta, J.: Zero-density estimate of L-functions attached to Maass forms. Acta Arith., 127(3), 273–284 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications Vol 1, Springer-Verlag, New York, 1985

    Google Scholar 

  6. Bump, D.: Automorphic Forms and Representations, Cambridge Stud. Adv. Math., 55, Cambridge University Press, Cambridge, 1997

    Google Scholar 

  7. Chandrasekharan, K., Narasimhan, R.: Functional equations with multiple gamma factors and the average order of arithmetical functions. Ann. of Math., 76(2), 93–136 (1962)

    Article  MathSciNet  Google Scholar 

  8. Gallagher, P. X.: A large sieve density estimate near σ = 1. Invent. Math., 11, 329–339 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  9. Pan, C. D., Pan, C. B.: Fundamentals of Analytic Number Theory (in Chinese), Science Press, Beijing, 1999

    Google Scholar 

  10. Gallagher, P. X.: The large sieve. Mathematika, 14, 14–20 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ramachandra, K.: Application of a theorem of Montgomery and Vaughan to the zeta-function. J. London Math. Soc., 10(2), 482–486 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kohnen, W., Sankaranarayanan, A., Sengupta, J.: The Quadratic Mean of Automorphic L-functions, Automorphic Forms and Zeta Functions, World Sci. Publ., Hackensack, 2006, 262–279

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Ji’nan, 250100, P. R. China

    Zhao Xu

Authors
  1. Zhao Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhao Xu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, Z. A new zero-density result of L-functions attached to Maass forms. Acta. Math. Sin.-English Ser. 27, 1149–1162 (2011). https://doi.org/10.1007/s10114-011-8310-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-011-8310-0

Keywords

MR(2000) Subject Classification

Search

Navigation