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]
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@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},
} -
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AUTHOR = {Deshouillers, J.-M. and Iwaniec, H.},
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author={Gradshteyn, I. S. and Ryzhik, I. M.},
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author={Hughes, C.},
NOTE={personal communication},
} -
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@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},
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MRCLASS = {10H05},
MRNUMBER = {0604215},
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@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},
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@book {Iwaniec2, MRKEY = {1942691},
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EDITION = {Second},
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PAGES = {xii+220},
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@book {IK, MRKEY = {2061214},
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TITLE = {Analytic {{N}}umber {{T}}heory},
SERIES = {Amer. Math. Soc. Colloq. Publ.},
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NOTE = {Ph.D. thesis, Rutgers University, 1998},
PUBLISHER = {ProQuest LLC, Ann Arbor, MI},
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ISBN = {978-0591-97579-6},
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MRNUMBER = {2697914},
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JOURNAL = {Invent. Math.},
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ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {11F66 (11F67 11M41)},
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@book {M, MRKEY = {1489236},
AUTHOR = {Motohashi, Yoichi},
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SERIES = {Cambridge Tracts in Math.},
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AUTHOR = {Ramachandra, K.},
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@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},
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ZBLNUMBER = {05233966},
} -
[S2]
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@article {S2, MRKEY = {1804529},
AUTHOR = {Soundararajan, K.},
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JOURNAL = {Ann. of Math.},
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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},
}