Skip to main content
SpringerLink
Log in

A uniform asymptotic formula for the second moment of primitive L-functions on the critical line

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We prove an asymptotic formula for the second moment of primitive L-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift, and twist.

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. O. Balkanova and D. Frolenkov, “Non-vanishing of automorphic L-functions of prime power level, ” arXiv: 1605.02434 [math.NT].

  2. H. M. Bui, “A note on the second moment of automorphic L-functions, ” Mathematika 56 (1), 35–44 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  3. V. A. Bykovskii, “A trace formula for the scalar product of Hecke series and its applications, ” Zap. Nauchn. Semin. POMI 226, 14–36 (1996) [J. Math. Sci. 89 (1), 915–932 (1998)].

    MathSciNet  MATH  Google Scholar 

  4. V. A. Bykovskii and D. A. Frolenkov, “On the second moment of L-series of holomorphic cusp forms on the critical line, ” Dokl. Akad. Nauk 463 (2), 133–136 (2015) [Dokl. Math. 92 (1), 417–420 (2015)].

    MathSciNet  MATH  Google Scholar 

  5. V. A. Bykovskii and D. A. Frolenkov, “Asymptotic formulas for the second moments of L-series associated to holomorphic cusp forms on the critical line, ” Izv. Ross. Akad. Nauk, Ser. Mat. (in press); arXiv: 1608.00555 [math.NT].

  6. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, HigherTranscendental Functions (McGraw-Hill, New York, 1953), Vol. 1, Bateman Manuscript Project.

    MATH  Google Scholar 

  7. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tablesof Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, Bateman Manuscript Project.

    MATH  Google Scholar 

  8. G. M. Fikhtengol’ts, A Course of Differential and Integral Calculus (Nauka, Moscow, 1970), Vol. 2 [in Russian].

    Google Scholar 

  9. D. A. Frolenkov, “On the uniform bounds on hypergeometric function, ” Dal’nevost. Mat. Zh. 15 (2), 289–298 (2015).

    MathSciNet  MATH  Google Scholar 

  10. I. S. Gradshteyn and I. M. Ryzhik, Tableof Integrals, Series, and Products, 6th ed. (Academic, San Diego, CA, 2000).

    MATH  Google Scholar 

  11. H. Iwaniec, Topicsin Classical Automorphic Forms (Am. Math. Soc., Providence, RI, 1997).

    MATH  Google Scholar 

  12. H. Iwaniec and P. Sarnak, “The non-vanishing of central values of automorphic L-functions and Landau–Siegel zeros, ” Isr. J. Math. 120, 155–177 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  13. NIST Handbook of Mathematical Functions, Ed. by F.W. J.Olver, D.W. Lozier, R.F. Boisvert, and C. W. Clarke (Cambridge Univ. Press, Cambridge, 2010).

  14. D. Rouymi, “Formules de trace et non-annulation de fonctions L automorphes au niveau pν, ” Acta Arith. 147, 1–32 (2011).

    Article  MathSciNet  Google Scholar 

  15. D. Rouymi, “Mollification et non annulation de fonctions L automorphes en niveau primaire, ” J. Number Theory 132 (1), 79–93 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Royer, “Sur les fonctions L de formes modulaires, ” PhD Thesis (Univ. Paris-Sud, Orsay, 2001).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Khabarovsk Division of the Institute of Applied Mathematics, Far Eastern Branch of the Russian Academy of Sciences, ul. Dzerzhinskogo 54, Khabarovsk, 680000, Russia

    Olga G. Balkanova

  2. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Dmitry A. Frolenkov

Authors
  1. Olga G. Balkanova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitry A. Frolenkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olga G. Balkanova.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 294, pp. 20–53.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Balkanova, O.G., Frolenkov, D.A. A uniform asymptotic formula for the second moment of primitive L-functions on the critical line. Proc. Steklov Inst. Math. 294, 13–46 (2016). https://doi.org/10.1134/S008154381606002X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S008154381606002X

Search

Navigation