Algebra & Number Theory

The mean value of symmetric square $L$-functions

Olga Balkanova and Dmitry Frolenkov

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the first moment of symmetric-square L -functions at the critical point in the weight aspect. Asymptotics with the best known error term O ( k 1 2 ) were obtained independently by Fomenko in 2003 and by Sun in 2013. We prove that there is an extra main term of size k 1 2 in the asymptotic formula and show that the remainder term decays exponentially in k . The twisted first moment was evaluated asymptotically by Ng with the error bounded by l k 1 2 + ϵ . We improve the error bound to l 5 6 + ϵ k 1 2 + ϵ unconditionally and to l 1 2 + ϵ k 1 2 under the Lindelöf hypothesis for quadratic Dirichlet L -functions.

Article information

Source
Algebra Number Theory, Volume 12, Number 1 (2018), 35-59.

Dates
Received: 20 October 2016
Revised: 30 June 2017
Accepted: 15 November 2017
First available in Project Euclid: 4 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1522807230

Digital Object Identifier
doi:10.2140/ant.2018.12.35

Mathematical Reviews number (MathSciNet)
MR3781432

Zentralblatt MATH identifier
06861735

Subjects
Primary: 11F12: Automorphic forms, one variable
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$ 34E05: Asymptotic expansions 34E20: Singular perturbations, turning point theory, WKB methods

Keywords
symmetric square $L$-functions weight aspect Gauss hypergeometric function Liouville–Green method WKB approximation

Citation

Balkanova, Olga; Frolenkov, Dmitry. The mean value of symmetric square $L$-functions. Algebra Number Theory 12 (2018), no. 1, 35--59. doi:10.2140/ant.2018.12.35. https://projecteuclid.org/euclid.ant/1522807230


Export citation

References

  • O. Balkanova and D. Frolenkov, “Moments of L-functions and the Liouville–Green method”, 2016.
  • H. Beitman and A. Erdelyi, Higher transcendental functions, McGraw-Hill, New York, 1953.
  • W. G. C. Boyd and T. M. Dunster, “Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions”, SIAM J. Math. Anal. 17:2 (1986), 422–450.
  • V. A. Bykovskiĭ, “Density theorems and the mean value of arithmetic functions on short intervals”, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. $($POMI$)$ 212:Anal. Teor. Chisel i Teor. Funktsiĭ. 12 (1994), 56–70, 196.
  • J. B. Conrey and H. Iwaniec, “The cubic moment of central values of automorphic $L$-functions”, Ann. of Math. $(2)$ 151:3 (2000), 1175–1216.
  • J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, “Integral moments of $L$-functions”, Proc. London Math. Soc. $(3)$ 91:1 (2005), 33–104.
  • A. Diaconu, D. Goldfeld, and J. Hoffstein, “Multiple Dirichlet series and moments of zeta and $L$-functions”, Compositio Math. 139:3 (2003), 297–360.
  • S. Farid Khwaja and A. B. Olde Daalhuis, “Uniform asymptotic expansions for hypergeometric functions with large parameters IV”, Anal. Appl. 12:6 (2014), 667–710.
  • O. M. Fomenko, “The behavior of automorphic $L$-functions at the points $s=1$ and $s=\frac 12$”, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. $($POMI$)$ 302:Anal. Teor. Chisel i Teor. Funkts. 19 (2003), 149–167, 201–202. In Russian; translated in Journal of Mathematical Sciences 129:3 (2005), 3898–3909.
  • S. Gelbart and H. Jacquet, “A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$”, Ann. Sci. École Norm. Sup. $(4)$ 11:4 (1978), 471–542.
  • I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007.
  • D. S. Jones, “Asymptotics of the hypergeometric function”, Math. Methods Appl. Sci. 24:6 (2001), 369–389.
  • R. Khan, “The first moment of the symmetric-square $L$-function”, J. Number Theory 124:2 (2007), 259–266.
  • R. Khan, “Non-vanishing of the symmetric square $L$-function at the central point”, Proc. Lond. Math. Soc. $(3)$ 100:3 (2010), 736–762.
  • S. F. Khwaja and A. B. O. Daalhuis, “Computation of the coefficients appearing in the uniform asymptotic expansions of integrals”, Stud. Appl. Math. 139:4 (2017), 551–567.
  • W. Kohnen and J. Sengupta, “On the average of central values of symmetric square $L$-functions in weight aspect”, Nagoya Math. J. 167 (2002), 95–100.
  • Y.-K. Lau, “Non-vanishing of symmetric square $L$-functions”, Proc. Amer. Math. Soc. 130:11 (2002), 3133–3139.
  • M. Lerch, “Note sur la fonction $\smash{{\mathfrak K} \left( {w,x,s} \right) = \sum_{k = 0}^\infty {\frac{{e^{2k\pi ix} }}{{\left( {w + k} \right)^s }}} }$”, Acta Math. 11:1-4 (1887), 19–24.
  • R. Lipschitz, “Untersuchung einer aus vier Elementen gebildeten Reihe”, J. Reine Angew. Math. 54 (1857), 313–328.
  • S. Liu, “The first moment of central values of symmetric square $L$-functions in the weight aspect”, Ramanujan J (online publication August 2017).
  • W. Luo, “Central values of the symmetric square $L$-functions”, Proc. Amer. Math. Soc. 140:10 (2012), 3313–3322.
  • M.-H. Ng, Moments of automorphic $L$-functions, Ph.D. thesis, University of Hong Kong, 2016.
  • M.-H. Ng, “The first moment of central values of symmetric square $L$-functions of cusp forms”, Acta Arith. 177:3 (2017), 277–291.
  • F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1974.
  • F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (editors), NIST handbook of mathematical functions, Cambridge University Press, Cambridge, 2010.
  • H. Petersson, “Über die Entwicklungskoeffizienten der automorphen Formen”, Acta Math. 58:1 (1932), 169–215.
  • G. Shimura, “On the holomorphy of certain Dirichlet series”, Proc. London Math. Soc. $(3)$ 31:1 (1975), 79–98.
  • K. Soundararajan and M. P. Young, “The prime geodesic theorem”, J. Reine Angew. Math. 676 (2013), 105–120.
  • Q. Sun, “On the first moment of the symmetric-square $L$-function”, Proc. Amer. Math. Soc. 141:2 (2013), 369–375.
  • A. Weil, “On some exponential sums”, Proc. Nat. Acad. Sci. U. S. A. 34 (1948), 204–207.
  • D. Zagier, “Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields”, pp. 105–169 in Modular functions of one variable, VI (Proc. Second Internat. Conf.) (Univ. Bonn, Bonn, 1976), edited by J.-P. Serre and D. B. Zagier, Lecture Notes in Math. 627, Springer, Berlin, Heidelberg, 1977.
  • N. Zavorotny, “Automorphic functions and number theory, part I, II”, Akad. Nauk SSSR, Dal'nevostochn. Otdel., Vladivostok 254 (1989), 69–124a. In Russian.