The fourth moment of Dirichlet $L$-functions

Pages 1-50 from Volume 173 (2011), Issue 1 by Matthew P. Young

Abstract

We compute the fourth moment of Dirichlet $L$-functions at the central point for prime moduli, with a power savings in the error term.

  • [CFKRS] Go to document 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., vol. 91, iss. 1, pp. 33-104, 2005.
    @article {CFKRS, MRKEY = {2149530},
      AUTHOR = {Conrey, J. B. and Farmer, D. W. and Keating, J. P. and Rubinstein, M. O. and Snaith, N. C.},
      TITLE = {Integral moments of {$L$}-functions},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {91},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {33--104},
      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {11M26},
      MRNUMBER = {2149530},
      MRREVIEWER = {K. Soundararajan},
      DOI = {10.1112/S0024611504015175},
      ZBLNUMBER = {1075.11058},
      }
  • [Deligne] Go to document P. Deligne, "La conjecture de Weil. I," Inst. Hautes Études Sci. Publ. Math., vol. 43, pp. 273-307, 1974.
    @article {Deligne, MRKEY = {0340258},
      AUTHOR = {Deligne, Pierre},
      TITLE = {La conjecture de {W}eil. {I}},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {43},
      YEAR = {1974},
      PAGES = {273--307},
      ISSN = {0073-8301},
      MRCLASS = {14G13},
      MRNUMBER = {0340258},
      MRREVIEWER = {Nicholas M. Katz},
      URL = {http://www.numdam.org/item?id=PMIHES_1974__43__273_0},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0287.14001},
      DOI={10.1007/BF02684373},
      }
  • [DI] Go to document J. -M. Deshouillers and H. Iwaniec, "Kloosterman sums and Fourier coefficients of cusp forms," Invent. Math., vol. 70, iss. 2, pp. 219-288, 1982/83.
    @article {DI, MRKEY = {684172},
      AUTHOR = {Deshouillers, J.-M. and Iwaniec, H.},
      TITLE = {Kloosterman sums and {F}ourier coefficients of cusp forms},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {70},
      YEAR = {1982/83},
      NUMBER = {2},
      PAGES = {219--288},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {10D12 (10G10)},
      MRNUMBER = {0684172},
      MRREVIEWER = {Audrey A. Terras},
      DOI = {10.1007/BF01390728},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0502.10021},
      }
  • [DFI] Go to document W. Duke, J. Friedlander, and H. Iwaniec, "Bounds for automorphic $L$-functions," Invent. Math., vol. 112, iss. 1, pp. 1-8, 1993.
    @article {DFI, MRKEY = {1207474},
      AUTHOR = {Duke, W. and Friedlander, J. 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},
      }
  • [Fouvry] Go to document &. Fouvry, "Sur le problème des diviseurs de Titchmarsh," J. Reine Angew. Math., vol. 357, pp. 51-76, 1985.
    @article {Fouvry, MRKEY = {783533},
      AUTHOR = {Fouvry, {É}tienne},
      TITLE = {Sur le problème des diviseurs de {T}itchmarsh},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {357},
      YEAR = {1985},
      PAGES = {51--76},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11N35},
      MRNUMBER = {0783533},
      MRREVIEWER = {H.-E. Richert},
      DOI = {10.1515/crll.1985.357.51},
      ZBLNUMBER = {0547.10039},
      }
  • [FI] &. Fouvry and H. Iwaniec, "The divisor function over arithmetic progressions," Acta Arith., vol. 61, iss. 3, pp. 271-287, 1992.
    @article {FI, MRKEY = {1161479},
      AUTHOR = {Fouvry, {É}tienne and Iwaniec, Henryk},
      TITLE = {The divisor function over arithmetic progressions},
      NOTE = {with an appendix by Nicholas Katz},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Acta Arithmetica},
      VOLUME = {61},
      YEAR = {1992},
      NUMBER = {3},
      PAGES = {271--287},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11L05 (11N56 11N64)},
      MRNUMBER = {1161479},
      MRREVIEWER = {Claudia A. Spiro-Silverman},
      ZBLNUMBER = {0764.11040},
      }
  • [FriedlanderI] Go to document J. B. Friedlander and H. Iwaniec, "Incomplete Kloosterman sums and a divisor problem," Ann. of Math., vol. 121, iss. 2, pp. 319-350, 1985.
    @article {FriedlanderI, MRKEY = {786351},
      AUTHOR = {Friedlander, John B. and Iwaniec, Henryk},
      TITLE = {Incomplete {K}loosterman sums and a divisor problem},
      NOTE = {with an appendix by Bryan J. Birch and Enrico Bombieri},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {121},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {319--350},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11N25 (11L40 11N69)},
      MRNUMBER = {0786351},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.2307/1971175},
      ZBLNUMBER = {0572.10029},
      }
  • [GR] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, New York: Academic Press, 1965.
    @book{GR,
      author={Gradshteyn, I. S. and Ryzhik, I. M.},
      TITLE={Table of {{I}}ntegrals, {{S}}eries, and {{P}}roducts},
      PUBLISHER={Academic Press},
      ADDRESS={New York},
      YEAR={1965},
      MRNUMBER={0197789},
      }
  • [H-B1] D. R. Heath-Brown, "The fourth power mean of Dirichlet’s $L$-functions," Analysis, vol. 1, iss. 1, pp. 25-32, 1981.
    @article {H-B1, MRKEY = {623640},
      AUTHOR = {Heath-Brown, D. R.},
      TITLE = {The fourth power mean of {D}irichlet's {$L$}-functions},
      JOURNAL = {Analysis},
      FJOURNAL = {Analysis. International Journal of Analysis and its Application},
      VOLUME = {1},
      YEAR = {1981},
      NUMBER = {1},
      PAGES = {25--32},
      ISSN = {0174-4747},
      MRCLASS = {10H08},
      MRNUMBER = {0623640},
      MRREVIEWER = {Matti Jutila},
      ZBLNUMBER = {0479.10027},
      }
  • [H-B2] Go to document D. R. Heath-Brown, "The fourth power moment of the Riemann zeta function," Proc. London Math. Soc., vol. 38, iss. 3, pp. 385-422, 1979.
    @article {H-B2, MRKEY = {532980},
      AUTHOR = {Heath-Brown, D. R.},
      TITLE = {The fourth power moment of the {R}iemann zeta function},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {38},
      YEAR = {1979},
      NUMBER = {3},
      PAGES = {385--422},
      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {10H05},
      MRNUMBER = {0532980},
      MRREVIEWER = {H. M. Stark},
      DOI = {10.1112/plms/s3-38.3.385},
      ZBLNUMBER = {0403.10018},
      }
  • [H-B3] 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 {H-B3, 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 = {0517737},
      MRREVIEWER = {M. I. Israilov},
      ZBLNUMBER = {0394.10020},
      }
  • [H] C. Hughes.
    @misc{H,
      author={Hughes, C.},
      NOTE={personal communication},
      }
  • [Ingham] A. E. Ingham, "Mean-value theorems in the theory of the Riemann zeta-function," Proc. London Math. Soc., vol. 27, pp. 273-300, 1926.
    @article{Ingham,
      author={Ingham, A. E.},
      TITLE={Mean-value theorems in the theory of the Riemann zeta-function},
      JOURNAL={Proc. London Math. Soc.},
      VOLUME={27},
      YEAR={1926},
      PAGES={273--300},
      }
  • [Iwaniec] H. Iwaniec, "Fourier coefficients of cusp forms and the Riemann zeta-function," in Seminar on Number Theory, 1979–1980 (French), Talence: Univ. Bordeaux I, 1980, p. x.
    @incollection {Iwaniec, MRKEY = {604215},
      AUTHOR = {Iwaniec, Henryk},
      TITLE = {Fourier coefficients of cusp forms and the {R}iemann zeta-function},
      BOOKTITLE = {Seminar on {N}umber {T}heory, 1979--1980 ({F}rench)},
      PAGES = {Exp. No. 18, 36},
      PUBLISHER = {Univ. Bordeaux I},
      ADDRESS = {Talence},
      YEAR = {1980},
      MRCLASS = {10H05},
      MRNUMBER = {0604215},
      MRREVIEWER = {R. A. Smith},
      ZBLNUMBER={0438.10030},
      }
  • [IwaniecMean] Go to document H. Iwaniec, "Mean values for Fourier coefficients of cusp forms and sums of Kloosterman sums," in Number Theory Days, 1980, Cambridge: Cambridge Univ. Press, 1982, vol. 56, pp. 306-321.
    @incollection {IwaniecMean, MRKEY = {697274},
      AUTHOR = {Iwaniec, Henryk},
      TITLE = {Mean values for {F}ourier coefficients of cusp forms and sums of {K}loosterman sums},
      BOOKTITLE = {Number Theory Days, 1980},
      VENUE={{E}xeter, 1980},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {56},
      PAGES = {306--321},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1982},
      MRCLASS = {10D12 (10G10)},
      MRNUMBER = {0697274},
      MRREVIEWER = {D. R. Heath-Brown},
      ZBLNUMBER = {0494.10035},
      DOI={10.1017/CBO9780511662027.021},
      }
  • [Iwaniec2] H. Iwaniec, Spectral Methods of Automorphic Forms, Second ed., Providence, RI: Amer. Math. Soc., 2002.
    @book {Iwaniec2, MRKEY = {1942691},
      AUTHOR = {Iwaniec, Henryk},
      TITLE = {Spectral {{M}}ethods of {{A}}utomorphic {{F}}orms},
      SERIES = {Grad. Stud. Math.},
      NUMBER = {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 = {1071.11001},
      ZBLNUMBER = {1059.11001},
      }
  • [KS] Go to document H. H. Kim, "Functoriality for the exterior square of ${ GL}_4$ and the symmetric fourth of ${ GL}_2$," J. Amer. Math. Soc., vol. 16, iss. 1, pp. 139-183, 2003.
    @article {KS, MRKEY = {1937203},
      AUTHOR = {Kim, Henry H.},
      TITLE = {Functoriality for the exterior square of {${\rm GL}\sb 4$} and the symmetric fourth of {${\rm GL}\sb 2$}},
      NOTE = {with Appendix 1 by Dinakar Ramakrishnan and Appendix 2 by Kim and Peter Sarnak},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {16},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {139--183},
      ISSN = {0894-0347},
      MRCLASS = {11F70 (11R39 22E46)},
      MRNUMBER = {2061214},
      MRREVIEWER = {Mahdi Asgari},
      DOI = {10.1090/S0894-0347-02-00410-1},
      ZBLNUMBER = {1018.11024},
      }
  • [Kowalski] Go to document E. G. Kowalski, The Rank of the Jacobian of Modular Curves: Analytic Methods, ProQuest LLC, Ann Arbor, MI, 1998.
    @book {Kowalski, MRKEY = {2613422},
      AUTHOR = {Kowalski, Emmanuel Guillaume},
      TITLE = {The {{R}}ank of the {J}acobian of {{M}}odular {{C}}urves: {A}nalytic {{M}}ethods},
      NOTE = {Ph.D. thesis, Rutgers University, 1998},
      PUBLISHER = {ProQuest LLC, Ann Arbor, MI},
      YEAR = {1998},
      PAGES = {209},
      ISBN = {978-0591-97579-6},
      MRCLASS = {Thesis},
      MRNUMBER = {2697914},
      URL = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9900669},
      }
  • [KMVdK] Go to document E. Kowalski, P. Michel, and J. VanderKam, "Mollification of the fourth moment of automorphic $L$-functions and arithmetic applications," Invent. Math., vol. 142, iss. 1, pp. 95-151, 2000.
    @article {KMVdK, MRKEY = {1784797},
      AUTHOR = {Kowalski, E. and Michel, P. and VanderKam, J.},
      TITLE = {Mollification of the fourth moment of automorphic {$L$}-functions and arithmetic applications},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {142},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {95--151},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F66 (11F67 11M41)},
      MRNUMBER = {2001m:11080},
      MRREVIEWER = {John B. Friedlander},
      DOI = {10.1007/s002220000086},
      ZBLNUMBER = {1054.11026},
      }
  • [Motohashi1] Go to document Y. Motohashi, "The binary additive divisor problem," Ann. Sci. École Norm. Sup., vol. 27, iss. 5, pp. 529-572, 1994.
    @article {Motohashi1, MRKEY = {1296556},
      AUTHOR = {Motohashi, Y{ō}ichi},
      TITLE = {The binary additive divisor problem},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {27},
      YEAR = {1994},
      NUMBER = {5},
      PAGES = {529--572},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {11N37 (11F72 11N75)},
      MRNUMBER = {1296556},
      MRREVIEWER = {D. R. Heath-Brown},
      URL = {http://www.numdam.org/item?id=ASENS_1994_4_27_5_529_0},
      ZBLNUMBER = {0819.11038},
      }
  • [M] Y. Motohashi, Spectral Theory of the Riemann Zeta-Function, Cambridge: Cambridge Univ. Press, 1997, vol. 127.
    @book {M, MRKEY = {1489236},
      AUTHOR = {Motohashi, Yoichi},
      TITLE = {Spectral {{T}}heory of the {R}iemann {Z}eta-{{F}}unction},
      SERIES = {Cambridge Tracts in Math.},
      VOLUME = {127},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1997},
      PAGES = {x+228},
      ISBN = {0-521-44520-5},
      MRCLASS = {11M06 (11F72)},
      MRNUMBER = {1489236},
      MRREVIEWER = {Don Redmond},
      ZBLNUMBER = {0878.11001},
      }
  • [R] Go to document K. Ramachandra, "Some remarks on a theorem of Montgomery and Vaughan," J. Number Theory, vol. 11, iss. 3 S. Chowla Anniversary Issue, pp. 465-471, 1979.
    @article {R, MRKEY = {544266},
      AUTHOR = {Ramachandra, K.},
      TITLE = {Some remarks on a theorem of {M}ontgomery and {V}aughan},
      JOURNAL = {J. Number Theory},
      FJOURNAL = {Journal of Number Theory},
      VOLUME = {11},
      YEAR = {1979},
      NUMBER = {3 S. Chowla Anniversary Issue},
      PAGES = {465--471},
      ISSN = {0022-314X},
      CODEN = {JNUTA9},
      MRCLASS = {10H99 (10H08)},
      MRNUMBER = {544266},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1016/0022-314X(79)90011-8},
      ZBLNUMBER = {0408.10028},
      }
  • [S] K. Soundararajan, "The fourth moment of Dirichlet $L$-functions," in Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2007, vol. 7, pp. 239-246.
    @incollection {S, MRKEY = {2362204},
      AUTHOR = {Soundararajan, K.},
      TITLE = {The fourth moment of {D}irichlet {$L$}-functions},
      BOOKTITLE = {Analytic Number Theory},
      SERIES = {Clay Math. Proc.},
      VOLUME = {7},
      PAGES = {239--246},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2007},
      MRCLASS = {11M06},
      MRNUMBER = {2362204},
      MRREVIEWER = {Arnaud Chadozeau},
      ZBLNUMBER = {05233966},
      }
  • [S2] Go to document K. Soundararajan, "Nonvanishing of quadratic Dirichlet $L$-functions at $s=\frac12$," Ann. of Math., vol. 152, iss. 2, pp. 447-488, 2000.
    @article {S2, MRKEY = {1804529},
      AUTHOR = {Soundararajan, K.},
      TITLE = {Nonvanishing of quadratic {D}irichlet {$L$}-functions at {$s=\frac12$}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {152},
      YEAR = {2000},
      NUMBER = {2},
      PAGES = {447--488},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11M20 (11R42)},
      MRNUMBER = {1804529},
      MRREVIEWER = {J. B. Conrey},
      DOI = {10.2307/2661390},
      ZBLNUMBER = {0964.11034},
      }

Keywords
, , , , ,
Mathematical Subject Classification
Primary: Secondary: ,
DOI
http://doi.org/10.4007/annals.2011.173.1.1
MR
2753598
Zbl
05960654
Milestones
Received: 22 September 2006
Accepted: 6 April 2010
Published online: 4 January 2011

Authors

Matthew P. Young

American Institute of Mathematics
Palo Alto, CA

Current address:

Texas A&M University
College Station, TX 77843-3368