Bounds for $\mathrm{GL}(3) \times \mathrm{GL}(2)$ $L$-functions and $\mathrm{GL}(3)$ $L$-functions

Abstract

In this paper, we will give the subconvexity bounds for self-dual $\mathrm{GL}(3)$ $L$-functions in the $t$ aspect as well as subconvexity bounds for self-dual $\mathrm{GL}(3)\times \mathrm{GL}(2)$ $L$-functions in the $\mathrm{GL}(2)$ spectral aspect.

  • [Bu] Go to document D. A. Burgess, "On character sums and $L$-series. II," Proc. London Math. Soc., vol. 13, pp. 524-536, 1963.
    @article {Bu, MRKEY = {0148626},
      AUTHOR = {Burgess, D. A.},
      TITLE = {On character sums and {$L$}-series. {{{\rm II}}}},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {13},
      YEAR = {1963},
      PAGES = {524--536},
      ISSN = {0024-6115},
      MRCLASS = {10.41},
      MRNUMBER = {0148626},
      MRREVIEWER = {L. Carlitz},
      DOI = {10.1112/plms/s3-13.1.524},
      ZBLNUMBER = {0123.04404},
      }
  • [BR] Go to document J. Bernstein and A. Reznikov, "Periods, subconvexity of $L$-functions and representation theory," J. Differential Geom., vol. 70, iss. 1, pp. 129-141, 2005.
    @article {BR, MRKEY = {2192063},
      AUTHOR = {Bernstein, Joseph and Reznikov, Andre},
      TITLE = {Periods, subconvexity of {$L$}-functions and representation theory},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {70},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {129--141},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {11F67 (11F70 22E45)},
      MRNUMBER = {2192063},
      MRREVIEWER = {Dipendra Prasad},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1143572016},
      ZBLNUMBER = {1097.11022},
      }
  • [CI] Go to document J. B. Conrey and H. Iwaniec, "The cubic moment of central values of automorphic $L$-functions," Ann. of Math., vol. 151, iss. 3, pp. 1175-1216, 2000.
    @article {CI, MRKEY = {1779567},
      AUTHOR = {Conrey, J. B. and Iwaniec, H.},
      TITLE = {The cubic moment of central values of automorphic {$L$}-functions},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {151},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {1175--1216},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11F66 (11F25 11L05 11M06)},
      MRNUMBER = {1779567},
      MRREVIEWER = {John B. Friedlander},
      DOI = {10.2307/121132},
      ZBLNUMBER = {0973.11056},
      }
  • [De] P. Deligne, Formes Modulaires et Repr\’présentations $l-$adiques, New York: Springer-Verlag, 1971.
    @book{De,
      author = {Deligne, Peter},
      TITLE = {Formes {{M}}odulaires et {{R}}epr\'présentations $l-$adiques},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {179},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1971},
      PAGES = {139--172},
      }
  • [DFI1] Go to document W. Duke, J. B. Friedlander, and H. Iwaniec, "Bounds for automorphic $L$-functions," Invent. Math., vol. 112, iss. 1, pp. 1-8, 1993.
    @article {DFI1, MRKEY = {1207474},
      AUTHOR = {Duke, W. and Friedlander, J. B. and Iwaniec, H.},
      TITLE = {Bounds for automorphic {$L$}-functions},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {112},
      YEAR = {1993},
      NUMBER = {1},
      PAGES = {1--8},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F66 (11F11)},
      MRNUMBER = {1207474},
      MRREVIEWER = {George Gilbert},
      DOI = {10.1007/BF01232422},
      ZBLNUMBER = {0765.11038},
      }
  • [DFI2] Go to document W. Duke, J. B. Friedlander, and H. Iwaniec, "Bounds for automorphic $L$-functions. II," Invent. Math., vol. 115, iss. 2, pp. 219-239, 1994.
    @article {DFI2, MRKEY = {1258904},
      AUTHOR = {Duke, W. and Friedlander, J. B. and Iwaniec, H.},
      TITLE = {Bounds for automorphic {$L$}-functions. {II}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {115},
      YEAR = {1994},
      NUMBER = {2},
      PAGES = {219--239},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F66 (11F11)},
      MRNUMBER = {1258904},
      MRREVIEWER = {George Gilbert},
      DOI = {10.1007/BF01231759},
      ZBLNUMBER = {0812.11032},
      }
  • [DFI3] Go to document W. Duke, J. B. Friedlander, and H. Iwaniec, "The subconvexity problem for Artin $L$-functions," Invent. Math., vol. 149, iss. 3, pp. 489-577, 2002.
    @article {DFI3, MRKEY = {1923476},
      AUTHOR = {Duke, W. and Friedlander, J. B. and Iwaniec, H.},
      TITLE = {The subconvexity problem for {A}rtin {$L$}-functions},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {149},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {489--577},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F66 (11F30 11M41 11R29)},
      MRNUMBER = {1923476},
      MRREVIEWER = {K. Soundararajan},
      DOI = {10.1007/s002220200223},
      ZBLNUMBER = {0099.37501},
      }
  • [Gol] Go to document D. Goldfeld, Automorphic Forms and $L$-Functions for the Group ${ GL}(n,\Bbb R)$, Cambridge: Cambridge Univ. Press, 2006, vol. 99.
    @book {Gol, MRKEY = {2254662},
      AUTHOR = {Goldfeld, Dorian},
      TITLE = {Automorphic {{F}}orms and {$L$}-{{F}}unctions for the {{G}}roup {${\rm GL}(n,\bold R)$}},
      SERIES = {Cambridge Stud. Adv. Math.},
      VOLUME = {99},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2006},
      PAGES = {xiv+493},
      ISBN = {978-0-521-83771-2; 0-521-83771-5},
      MRCLASS = {11F55 (11F66 11F70 11R39)},
      MRNUMBER = {2254662},
      MRREVIEWER = {Emmanuel P. Royer},
      ZBLNUMBER = {1108.11039},
      DOI={10.1017/CBO9780511542923},
      }
  • [GL] Go to document D. Goldfeld and X. Li, "Voronoi formulas on ${ GL}(n)$," Int. Math. Res. Not., vol. 2006, p. I, 2006.
    @article {GL, MRKEY = {2233713},
      AUTHOR = {Goldfeld, Dorian and Li, Xiaoqing},
      TITLE = {Voronoi formulas on {${\rm GL}(n)$}},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2006},
      PAGES = {Art. ID 86295, 25},
      ISSN = {1073-7928},
      MRCLASS = {11F66 (11F70 11M41)},
      MRNUMBER = {2233713},
      MRREVIEWER = {Kazuyuki Hatada},
      DOI = {10.1155/IMRN/2006/86295},
      VOLUME = {2006},
      }
  • [Go] Go to document A. Good, "The square mean of Dirichlet series associated with cusp forms," Mathematika, vol. 29, iss. 2, pp. 278-295 (1983), 1982.
    @article {Go, MRKEY = {696884},
      AUTHOR = {Good, Anton},
      TITLE = {The square mean of {D}irichlet series associated with cusp forms},
      JOURNAL = {Mathematika},
      FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
      VOLUME = {29},
      YEAR = {1982},
      NUMBER = {2},
      PAGES = {278--295 (1983)},
      ISSN = {0025-5793},
      CODEN = {MTKAAB},
      MRCLASS = {10D24 (10D12 10H10)},
      MRNUMBER = {696884},
      MRREVIEWER = {J{ü}rgen Elstrodt},
      DOI = {10.1112/S0025579300012377},
      ZBLNUMBER = {0497.10016},
      }
  • [GR] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Sixth ed., San Diego, CA: Academic Press, 2000.
    @book {GR, MRKEY = {1773820},
      AUTHOR = {Gradshteyn, I. S. and Ryzhik, I. M.},
      TITLE = {Table of {{I}}ntegrals, {{S}}eries, and {{P}}roducts},
      EDITION = {Sixth},
      NOTE = {translated from the Russian},
      PUBLISHER = {Academic Press},
      ADDRESS = {San Diego, CA},
      YEAR = {2000},
      PAGES = {xlvii+1163},
      ISBN = {0-12-294757-6},
      MRCLASS = {00A22 (33-00 65-00 65A05)},
      MRNUMBER = {1773820},
      ZBLNUMBER = {0981.65001},
      }
  • [HM] Go to document G. Harcos and P. Michel, "The subconvexity problem for Rankin-Selberg $L$-functions and equidistribution of Heegner points. II," Invent. Math., vol. 163, iss. 3, pp. 581-655, 2006.
    @article {HM, MRKEY = {2207235},
      AUTHOR = {Harcos, Gergely and Michel, Philippe},
      TITLE = {The subconvexity problem for {R}ankin-{S}elberg {$L$}-functions and equidistribution of {H}eegner points. {II}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {163},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {581--655},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F66 (11F67 11M41)},
      MRNUMBER = {2207235},
      MRREVIEWER = {K. Soundararajan},
      DOI = {10.1007/s00222-005-0468-6},
      ZBLNUMBER = {1111.11027},
      }
  • [He] Go to document D. R. Heath-Brown, "The twelfth power moment of the Riemann-function," Quart. J. Math. Oxford Ser., vol. 29, iss. 116, pp. 443-462, 1978.
    @article {He, MRKEY = {517737},
      AUTHOR = {Heath-Brown, D. R.},
      TITLE = {The twelfth power moment of the {R}iemann-function},
      JOURNAL = {Quart. J. Math. Oxford Ser.},
      FJOURNAL = {The Quarterly Journal of Mathematics. Oxford. Second Series},
      VOLUME = {29},
      YEAR = {1978},
      NUMBER = {116},
      PAGES = {443--462},
      ISSN = {0033-5606},
      CODEN = {QJMAAT},
      MRCLASS = {10H05},
      MRNUMBER = {517737},
      MRREVIEWER = {M. I. Israilov},
      ZBLNUMBER = {0394.10020},
      DOI={10.1093/qmath/29.4.443},
      }
  • [HL94] Go to document J. Hoffstein and P. Lockhart, "Coefficients of Maass forms and the Siegel zero," Ann. of Math., vol. 140, pp. 161-181, 1994.
    @article{HL94,
      author={Hoffstein, J. and Lockhart, P.},
      TITLE={Coefficients of {M}aass forms and the {S}iegel zero},
      NOTE={with an appendix by D. Goldfeld, J. Hoffstein, and D. Lieman},
      JOURNAL={Ann. of Math.},
      VOLUME={140},
      YEAR={1994},
      PAGES={161--181},
      MRNUMBER={1289494},
      ZBLNUMBER={0814.11032},
      DOI={10.2307/2118543},
      }
  • [Hu] M. N. Huxley, Area, Lattice Points, and Exponential Sums, New York: The Clarendon Press Oxford University Press, 1996, vol. 13.
    @book {Hu, MRKEY = {1420620},
      AUTHOR = {Huxley, M. N.},
      TITLE = {Area, {{L}}attice {{P}}oints, and {{E}}xponential {{S}}ums},
      SERIES = {London Math. Soc. Monogr. New Series},
      VOLUME = {13},
      NOTE = {Oxford Science Publications},
      PUBLISHER = {The Clarendon Press Oxford University Press},
      ADDRESS = {New York},
      YEAR = {1996},
      PAGES = {xii+494},
      ISBN = {0-19-853466-3},
      MRCLASS = {11L07 (11J54 11P21)},
      MRNUMBER = {1420620},
      MRREVIEWER = {R. C. Baker},
      ZBLNUMBER = {0861.11002},
      }
  • [Iv] Go to document A. Ivić, "On the ternary additive divisor problem and the sixth moment of the zeta-function," in Sieve Methods, Exponential Sums, and their Applications in Number Theory, Cambridge: Cambridge Univ. Press, 1997, vol. 237, pp. 205-243.
    @incollection {Iv, MRKEY = {1635762},
      AUTHOR = {Ivi{ć},
      Aleksandar},
      TITLE = {On the ternary additive divisor problem and the sixth moment of the zeta-function},
      BOOKTITLE = {Sieve {{M}}ethods, {{E}}xponential {{S}}ums, and their {{A}}pplications in {{N}}umber {{T}}heory},
      VENUE={{C}ardiff, 1995},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {237},
      PAGES = {205--243},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1997},
      MRCLASS = {11M06 (11L07 11N37)},
      MRNUMBER = {1635762},
      MRREVIEWER = {Lutz Gerhard Lucht},
      ZBLNUMBER = {0924.11069},
      DOI={10.1017/CBO9780511526091.015},
      }
  • [Iw1] H. Iwaniec, Topics in Classical Automorphic Forms, Providence, RI: Amer. Math. Soc., 1997, vol. 17.
    @book {Iw1, MRKEY = {1474964},
      AUTHOR = {Iwaniec, Henryk},
      TITLE = {Topics in {{C}}lassical {{A}}utomorphic {{F}}orms},
      SERIES = {Grad. Stud. Math.},
      VOLUME = {17},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1997},
      PAGES = {xii+259},
      ISBN = {0-8218-0777-3},
      MRCLASS = {11Fxx (11-02)},
      MRNUMBER = {1474964},
      MRREVIEWER = {B. Ramakrishnan},
      ZBLNUMBER = {0905.11023},
      }
  • [Iw2] H. Iwaniec, Spectral Methods of Automorphic Forms, Second ed., Providence, RI: Amer. Math. Soc., 2002, vol. 53.
    @book {Iw2, MRKEY = {1942691},
      AUTHOR = {Iwaniec, Henryk},
      TITLE = {Spectral {{M}}ethods of {{A}}utomorphic {{F}}orms},
      SERIES = {Grad. Stud. Math.},
      VOLUME= {53},
      EDITION = {Second},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2002},
      PAGES = {xii+220},
      ISBN = {0-8218-3160-7},
      MRCLASS = {11F72 (11F12 11F37)},
      MRNUMBER = {1942691},
      ZBLNUMBER = {1006.11024},
      }
  • [IK] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53.
    @book {IK, MRKEY = {2061214},
      AUTHOR = {Iwaniec, Henryk and Kowalski, Emmanuel},
      TITLE = {Analytic {{N}}umber {{T}}heory},
      SERIES = {Amer. Math. Soc. Colloq. Publ.},
      VOLUME = {53},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2004},
      PAGES = {xii+615},
      ISBN = {0-8218-3633-1},
      MRCLASS = {11-02 (11Fxx 11Lxx 11Mxx 11Nxx)},
      MRNUMBER = {2061214},
      MRREVIEWER = {K. Soundararajan},
      ZBLNUMBER = {1059.11001},
      }
  • [ILS] Go to document H. Iwaniec, W. Luo, and P. Sarnak, "Low lying zeros of families of $L$-functions," Inst. Hautes Études Sci. Publ. Math., iss. 91, pp. 55-131 (2001), 2000.
    @article {ILS, MRKEY = {1828743},
      AUTHOR = {Iwaniec, Henryk and Luo, Wenzhi and Sarnak, Peter},
      TITLE = {Low lying zeros of families of {$L$}-functions},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {91},
      YEAR = {2000},
      PAGES = {55--131 (2001)},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {11M41 (11F66 11M26)},
      MRNUMBER = {1828743},
      MRREVIEWER = {Philippe G. Michel},
      URL = {http://www.numdam.org/item?id=PMIHES_2000__91__55_0},
      ZBLNUMBER = {1012.11041},
      DOI={10.1007/BF02698741},
      }
  • [JS] H. Jacquet and J. Shalika, "Exterior square $L$-functions," in Automorphic Forms, Shimura Varieties, and $L$-Functions, Vol. II, Boston, MA: Academic Press, 1990, vol. 11, pp. 143-226.
    @incollection {JS, MRKEY = {1044830},
      AUTHOR = {Jacquet, Herv{é} and Shalika, Joseph},
      TITLE = {Exterior square {$L$}-functions},
      BOOKTITLE = {Automorphic {{F}}orms, {S}himura {V}arieties, and {$L$}-{{F}}unctions, {V}ol. {\rm II}},
      VENUE={{A}nn {A}rbor, {MI},
      1988},
      SERIES = {Perspect. Math.},
      VOLUME= {11},
      PAGES = {143--226},
      PUBLISHER = {Academic Press},
      ADDRESS = {Boston, MA},
      YEAR = {1990},
      MRCLASS = {11F67 (11F27 11F70 22E50)},
      MRNUMBER = {91g:11050},
      MRREVIEWER = {Stephen Gelbart},
      ZBLNUMBER = {0695.10025},
      }
  • [La] Go to document E. M. Lapid, "On the nonnegativity of Rankin-Selberg $L$-functions at the center of symmetry," Int. Math. Res. Not., vol. 2003, iss. 2, pp. 65-75, 2003.
    @article {La, MRKEY = {1936579},
      AUTHOR = {Lapid, Erez M.},
      TITLE = {On the nonnegativity of {R}ankin-{S}elberg {$L$}-functions at the center of symmetry},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {65--75},
      ISSN = {1073-7928},
      MRCLASS = {11F70 (11F67)},
      MRNUMBER = {1936579},
      MRREVIEWER = {Henry H. Kim},
      DOI = {10.1155/S1073792803204013},
      VOLUME = {2003},
      ZBLNUMBER = {1046.11032},
      }
  • [LYL] Go to document Y. Lau, J. Liu, and Y. Ye, "A new bound $k^{2/3+\epsilon}$ for Rankin-Selberg $L$-functions for Hecke congruence subgroups," IMRP Int. Math. Res. Pap., p. I, 2006.
    @article {LYL, MRKEY = {2235495},
      AUTHOR = {Lau, Yuk-Kam and Liu, Jianya and Ye, Yangbo},
      TITLE = {A new bound {$k\sp {2/3+\epsilon}$} for {R}ankin-{S}elberg {$L$}-functions for {H}ecke congruence subgroups},
      JOURNAL = {{\rm IMRP} Int. Math. Res. Pap.},
      FJOURNAL = {IMRP. International Mathematics Research Papers},
      YEAR = {2006},
      PAGES = {Art. ID 35090, 78},
      ISSN = {1687-3017},
      MRCLASS = {11F66 (11F67 11F72 11M36)},
      MRNUMBER = {2235495},
      MRREVIEWER = {Valentin Blomer},
      ZBLNUMBER={1178.11041},
      DOI={10.1155/IMRP/2006/35090},
      }
  • [Li] Go to document X. Li, "The central value of the Rankin-Selberg $L$-functions," Geom. Funct. Anal., vol. 18, iss. 5, pp. 1660-1695, 2009.
    @article {Li, MRKEY = {2481739},
      AUTHOR = {Li, Xiaoqing},
      TITLE = {The central value of the {R}ankin-{S}elberg {$L$}-functions},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {18},
      YEAR = {2009},
      NUMBER = {5},
      PAGES = {1660--1695},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {11F67 (11F41)},
      MRNUMBER = {2481739},
      MRREVIEWER = {Nicolas P. Templier},
      DOI = {10.1007/s00039-008-0692-5},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {1188.11021},
      }
  • [LRS] W. Luo, Z. Rudnick, and P. Sarnak, "On the generalized Ramanujan conjecture for ${ GL}(n)$," in Automorphic Forms, Automorphic Representations, and Arithmetic, Providence, RI: Amer. Math. Soc., 1999, vol. 66, pp. 301-310.
    @incollection {LRS, MRKEY = {1703764},
      AUTHOR = {Luo, Wenzhi and Rudnick, Ze{é}v and Sarnak, Peter},
      TITLE = {On the generalized {R}amanujan conjecture for {${\rm GL}(n)$}},
      BOOKTITLE = {Automorphic {{F}}orms, {{A}}utomorphic {{R}}epresentations, and {{A}}rithmetic},
      VENUE={{F}ort {W}orth, {TX},
      1996},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME= {66},
      PAGES = {301--310},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1999},
      MRCLASS = {11F70 (11F72)},
      MRNUMBER = {1703764},
      MRREVIEWER = {Stefan K{ü}hnlein},
      ZBLNUMBER = {0965.11023},
      }
  • [KMV] Go to document E. Kowalski, P. Michel, and J. VanderKam, "Rankin-Selberg $L$-functions in the level aspect," Duke Math. J., vol. 114, iss. 1, pp. 123-191, 2002.
    @article {KMV, MRKEY = {1915038},
      AUTHOR = {Kowalski, E. and Michel, P. and VanderKam, J.},
      TITLE = {Rankin-{S}elberg {$L$}-functions in the level aspect},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {114},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {123--191},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {11F66 (11G40)},
      MRNUMBER = {1915038},
      MRREVIEWER = {K. Soundararajan},
      DOI = {10.1215/S0012-7094-02-11416-1},
      ZBLNUMBER = {1035.11018},
      }
  • [Me] T. Meurman, "On the order of the Maass $L$-function on the critical line," in Number Theory, Vol. I, Amsterdam: North-Holland, 1990, vol. 51, pp. 325-354.
    @incollection {Me, MRKEY = {1058223},
      AUTHOR = {Meurman, Tom},
      TITLE = {On the order of the {M}aass {$L$}-function on the critical line},
      BOOKTITLE = {Number {{T}}heory, {V}ol. {\rm I}},
      VENUE={{B}udapest, 1987},
      SERIES = {Colloq. Math. Soc. János Bolyai},
      VOLUME = {51},
      PAGES = {325--354},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {1990},
      MRCLASS = {11F66 (11F30 11M41)},
      MRNUMBER = {1058223},
      MRREVIEWER = {Kazuyuki Hatada},
      ZBLNUMBER = {0724.11029},
      }
  • [Mi] Go to document P. Michel, "The subconvexity problem for Rankin-Selberg $L$-functions and equidistribution of Heegner points," Ann. of Math., vol. 160, iss. 1, pp. 185-236, 2004.
    @article{Mi, MRKEY = {2119720},
      AUTHOR = {Michel, Philippe},
      TITLE = {The subconvexity problem for {R}ankin-{S}elberg {$L$}-functions and equidistribution of {H}eegner points},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {160},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {185--236},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11F66 (11G18)},
      MRNUMBER = {2119720},
      MRREVIEWER = {Gergely Harcos},
      DOI = {10.4007/annals.2004.160.185},
      ZBLNUMBER = {1068.11033},
      }
  • [MV1] P. Michel and A. Venkatesh, "Heegner points and nonvanishing of Rankin-Selberg $L$-functions," in Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2007, vol. 7, pp. 169-183.
    @incollection{MV1,
      author={Michel, Philippe and Venkatesh, Akshay},
      TITLE = {Heegner points and nonvanishing of Rankin-Selberg $L$-functions},
      BOOKTITLE={Analytic {{N}}umber {{T}}heory},
      SERIES={Clay Math. Proc.},
      VOLUME={7},
      PAGES={169--183},
      PUBLISHER={Amer. Math. Soc.},
      ADDRESS={Providence, RI},
      YEAR={2007},
      MRNUMBER={2362200},
      ZBLNUMBER={pre05233962},
      }
  • [MV2] Go to document P. Michel and A. Venkatesh, "Equidistribution, $L$-functions and ergodic theory: on some problems of Yu. Linnik," in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 421-457.
    @incollection {MV2, MRKEY = {2275604},
      AUTHOR = {Michel, Philippe and Venkatesh, Akshay},
      TITLE = {Equidistribution, {$L$}-functions and ergodic theory: on some problems of {Y}u. {L}innik},
      BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {{\rm II}}},
      PAGES = {421--457},
      PUBLISHER = {Eur. Math. Soc., Zürich},
      YEAR = {2006},
      MRCLASS = {11F66 (11F67 11M41 37A45)},
      MRNUMBER = {2275604},
      MRREVIEWER = {Gergely Harcos},
      ZBLNUMBER={1157.11019},
      DOI={10.4171/022-2/19},
      }
  • [Mil] Go to document S. D. Miller, "Cancellation in additively twisted sums on ${ GL}(n)$," Amer. J. Math., vol. 128, iss. 3, pp. 699-729, 2006.
    @article {Mil, MRKEY = {2230922},
      AUTHOR = {Miller, Stephen D.},
      TITLE = {Cancellation in additively twisted sums on {${\rm GL}(n)$}},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {128},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {699--729},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {11F67 (11F70)},
      MRNUMBER = {2230922},
      MRREVIEWER = {Emmanuel P. Royer},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v128/128.3miller.pdf},
      ZBLNUMBER = {1142.11033},
      DOI={10.1353/ajm.2006.0027},
      }
  • [MS] Go to document S. D. Miller and W. Schmid, "Automorphic distributions, $L$-functions, and Voronoi summation for ${ GL}(3)$," Ann. of Math., vol. 164, iss. 2, pp. 423-488, 2006.
    @article {MS, MRKEY = {2247965},
      AUTHOR = {Miller, Stephen D. and Schmid, Wilfried},
      TITLE = {Automorphic distributions, {$L$}-functions, and {V}oronoi summation for {${\rm GL}(3)$}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {164},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {423--488},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11F66 (11F70 11M41)},
      MRNUMBER = {2247965},
      MRREVIEWER = {Andre Reznikov},
      DOI = {10.4007/annals.2006.164.423},
      ZBLNUMBER = {1162.11341},
      }
  • [Sa] Go to document P. Sarnak, "Estimates for Rankin-Selberg $L$-functions and quantum unique ergodicity," J. Funct. Anal., vol. 184, iss. 2, pp. 419-453, 2001.
    @article {Sa, MRKEY = {1851004},
      AUTHOR = {Sarnak, Peter},
      TITLE = {Estimates for {R}ankin-{S}elberg {$L$}-functions and quantum unique ergodicity},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {184},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {419--453},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {11F72 (11F66 37A45)},
      MRNUMBER = {1851004},
      MRREVIEWER = {Henry H. Kim},
      DOI = {10.1006/jfan.2001.3783},
      ZBLNUMBER = {1006.11022},
      }
  • [So] D. Soudry, "On Langlands functoriality from classical groups to ${ GL}_n$. Automorphic forms. I," Astérisque, vol. 298, pp. 335-390, 2005.
    @article {So, MRKEY = {2141707},
      AUTHOR = {Soudry, David},
      TITLE = {On {L}anglands functoriality from classical groups to ${\rm GL}\sb n$. {A}utomorphic forms. {{I}}},
      JOURNAL = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {298},
      YEAR = {2005},
      PAGES = {335--390},
      ISSN = {0303-1179},
      MRCLASS = {11F70 (11R39 22E55)},
      MRNUMBER = {2141707},
      MRREVIEWER = {Mahdi Asgari},
      ZBLNUMBER = {1086.11025},
      }
  • [Ti] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Second ed., New York: The Clarendon Press Oxford University Press, 1986.
    @book {Ti, MRKEY = {882550},
      AUTHOR = {Titchmarsh, E. C.},
      TITLE = {The {{T}}heory of the {R}iemann {{Z}}eta-{{F}}unction},
      EDITION = {Second},
      NOTE = {edited and with a preface by D. R. Heath-Brown},
      PUBLISHER = {The Clarendon Press Oxford University Press},
      ADDRESS = {New York},
      YEAR = {1986},
      PAGES = {x+412},
      ISBN = {0-19-853369-1},
      MRCLASS = {11M06},
      MRNUMBER = {882550},
      MRREVIEWER = {Matti Jutila},
      ZBLNUMBER = {0601.10026},
      }
  • [Ve] Go to document A. Venkatesh, "Sparse equidistribution problems, period bounds, and subconvexity," Ann. of Math., vol. 172, pp. 989-1094, 2010.
    @article{Ve,
      author={Venkatesh, A.},
      TITLE={Sparse equidistribution problems, period bounds, and subconvexity},
      JOURNAL={Ann. of Math.},
      VOLUME={172},
      PAGES={989--1094},
      YEAR={2010},
      MRNUMBER={2680486},
      ZBLNUMBER={pre05808581},
      DOI={10.4007/annals.2010.172.989},
      }
  • [Wa] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge: Cambridge Univ. Press, 1995.
    @book {Wa, MRKEY = {1349110},
      AUTHOR = {Watson, G. N.},
      TITLE = {A {{T}}reatise on the {{T}}heory of {B}essel {{F}}unctions},
      SERIES = {Cambridge Mathematical Library},
      NOTE = {reprint of the second (1944) edition},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1995},
      PAGES = {viii+804},
      ISBN = {0-521-48391-3},
      MRCLASS = {33C10 (01A75 33-01 33-02)},
      MRNUMBER = {1349110},
      MRREVIEWER = {R. A. Askey},
      ZBLNUMBER = {0849.33001},
      }
  • [We] H. Weyl, "Zur abschätzung von $\zeta (1+it)$," Math. Z., vol. 10, pp. 88-101, 1921.
    @article{We,
      author={Weyl, H.},
      TITLE={Zur abschätzung von $\zeta (1+it)$},
      JOURNAL={Math. Z.},
      VOLUME={10},
      YEAR={1921},
      PAGES={88--101},
      JFMNUMBER={48.0346.01},
      }

Authors

Xiaoqing Li

Department of Mathematics
State University of New York at Buffalo
Buffalo, NY 14260-2900