Additive twists of Fourier coefficients of $GL(3)$ Maass forms
HTML articles powered by AMS MathViewer
- by Xiannan Li PDF
- Proc. Amer. Math. Soc. 142 (2014), 1825-1836 Request permission
Abstract:
We prove cancellation in a sum of Fourier coefficents of a $GL(3)$ form $F$ twisted by additive characters, uniformly in the form $F$. Previously, this type of result was available only when $F$ is a symmetric square lift.References
- Farrell Brumley, Second order average estimates on local data of cusp forms, Arch. Math. (Basel) 87 (2006), no.Β 1, 19β32. MR 2246403, DOI 10.1007/s00013-005-1632-3
- Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423
- Dorian Goldfeld, Automorphic forms and $L$-functions for the group $\textrm {GL}(n,\mathbf R)$, Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. MR 2254662, DOI 10.1017/CBO9780511542923
- Dorian Goldfeld and Xiaoqing Li, Voronoi formulas on $\textrm {GL}(n)$, Int. Math. Res. Not. , posted on (2006), Art. ID 86295, 25. MR 2233713, DOI 10.1155/IMRN/2006/86295
- G. H. Hardy and J. E. Littlewood, Some problems of diophantine approximation, Acta Math. 37 (1914), no.Β 1, 193β239. MR 1555099, DOI 10.1007/BF02401834
- M. N. Huxley, Area, lattice points, and exponential sums, London Mathematical Society Monographs. New Series, vol. 13, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1420620
- Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964, DOI 10.1090/gsm/017
- Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
- Xiannan Li, Upper bounds on $L$-functions at the edge of the critical strip, Int. Math. Res. Not. IMRN 4 (2010), 727β755. MR 2595006, DOI 10.1093/imrn/rnp148
- Xiaoqing Li and Matthew P. Young, Additive twists of Fourier coefficients of symmetric-square lifts, J. Number Theory 132 (2012), no.Β 7, 1626β1640. MR 2903173, DOI 10.1016/j.jnt.2011.12.017
- Stephen D. Miller, Cancellation in additively twisted sums on $\textrm {GL}(n)$, Amer. J. Math. 128 (2006), no.Β 3, 699β729. MR 2230922, DOI 10.1353/ajm.2006.0027
- Stephen D. Miller and Wilfried Schmid, Automorphic distributions, $L$-functions, and Voronoi summation for $\textrm {GL}(3)$, Ann. of Math. (2) 164 (2006), no.Β 2, 423β488. MR 2247965, DOI 10.4007/annals.2006.164.423
- H. L. Montgomery and R. C. Vaughan, Exponential sums with multiplicative coefficients, Invent. Math. 43 (1977), no.Β 1, 69β82. MR 457371, DOI 10.1007/BF01390204
- E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
Additional Information
- Xiannan Li
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
- Address at time of publication: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
- MR Author ID: 867056
- Received by editor(s): February 21, 2012
- Received by editor(s) in revised form: June 26, 2012
- Published electronically: February 24, 2014
- Communicated by: Matthew A. Papanikolas
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 1825-1836
- MSC (2010): Primary 11F03, 11M41
- DOI: https://doi.org/10.1090/S0002-9939-2014-11909-5
- MathSciNet review: 3182004