Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 

 

Central values of the symmetric square $ L$-functions


Author: Wenzhi Luo
Journal: Proc. Amer. Math. Soc. 140 (2012), 3313-3322
MSC (2010): Primary 11F11, 11F66, 11F67
Published electronically: January 31, 2012
MathSciNet review: 2929002
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Abstract | References | Similar Articles | Additional Information

Abstract: We establish a sharp bound for the square mean of the central values of the symmetric square $ L$-functions associated to holomorphic cusp forms of level $ 1$, as the weight $ k$ varies in the short interval $ [K,\; K + K^{1/2 + \epsilon }]$.


References [Enhancements On Off] (What's this?)

  • 1. Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of 𝐺𝐿(2) and 𝐺𝐿(3), Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. MR 533066
  • 2. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey, Academic Press, New York-London, 1965. MR 0197789
  • 3. H. Iwaniec and P. Michel, The second moment of the symmetric square 𝐿-functions, Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 2, 465–482. MR 1833252
  • 4. M. Jutila, On the mean value of 𝐿(1\over2,𝜒) for real characters, Analysis 1 (1981), no. 2, 149–161. MR 632705, 10.1524/anly.1981.1.2.149
  • 5. Rizwanur Khan, Non-vanishing of the symmetric square 𝐿-function at the central point, Proc. Lond. Math. Soc. (3) 100 (2010), no. 3, 736–762. MR 2640289, 10.1112/plms/pdp048
  • 6. Winfried Kohnen and Jyoti Sengupta, On the average of central values of symmetric square 𝐿-functions in weight aspect, Nagoya Math. J. 167 (2002), 95–100. MR 1924720
  • 7. Goro Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), no. 1, 79–98. MR 0382176
  • 8. D. Zagier, Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1977, pp. 105–169. Lecture Notes in Math., Vol. 627. MR 0485703

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Additional Information

Wenzhi Luo
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Email: wluo@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11174-8
Received by editor(s): February 10, 2011
Received by editor(s) in revised form: March 31, 2011
Published electronically: January 31, 2012
Additional Notes: The author’s research was partially supported by NSF grant DMS-0855600
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.