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Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms

Part of: Discontinuous groups and automorphic forms Multiplicative number theory

Published online by Cambridge University Press:  24 April 2014

Étienne Fouvry  and
Satadal Ganguly
Étienne Fouvry
Affiliation:
Laboratoire de Mathématique, Université Paris Sud, UMR 8628, CNRS, Orsay, F–91405, France email Etienne.Fouvry@math.u-psud.fr
Satadal Ganguly
Affiliation:
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata 700108, India email sgisical@gmail.com
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Abstract

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Let $\nu _{f}(n)$ be the $n\mathrm{th}$ normalized Fourier coefficient of a Hecke–Maass cusp form $f$ for ${\rm SL }(2,\mathbb{Z})$ and let $\alpha $ be a real number. We prove strong oscillations of the argument of $\nu _{f}(n)\mu (n) \exp (2\pi i n \alpha )$ as $n$ takes consecutive integral values.

Keywords

cusp formsMöbius functionadditive characters

MSC classification

Secondary: 11F30: Fourier coefficients of automorphic forms 11N75: Applications of automorphic functions and forms to multiplicative problems

Information

Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 5 , May 2014 , pp. 763 - 797
Copyright
© The Author(s) 2014 

References

Bourgain, J., Sarnak, P. and Ziegler, T., Disjointness of Mobius from horocycle flows, in From Fourier analysis and number theory to radon transforms and geometry, Developments in Mathematics, vol. 28 (Springer, New York, 2013), 67–83.Google Scholar
Brumley, F., Maass cusp forms with quadratic integer coefficients, Int. Math. Res. Not. IMRN 18 (2003), 983–997.Google Scholar
Bump, D., Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55 (Cambridge University Press, Cambridge, 1998).Google Scholar
Carleson, L., On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157.Google Scholar
Cellarosi, F. and Sinai, Y. G., Ergodic properties of square-free numbers, J. Eur. Math. Soc. (JEMS) 15 (2013), 1343–1374.Google Scholar
Conrey, B. and Iwaniec, H., The cubic moment of central values of automorphic L-functions, Ann. of Math. (2) 151 (2000), 1175–1216.CrossRefGoogle Scholar
Davenport, H., On some infinite series involving arithmetical functions. II, Q. J. Math. 8 (1937), 313–320.Google Scholar
Duke, W., Friedlander, J. B. and Iwaniec, H., The subconvexity problem for Artin L-functions, Invent. Math. 149 (2002), 489–577.Google Scholar
Elliott, P. D. T. A., Multiplicative functions and Ramanujan’s τ-function, J. Aust. Math. Soc., Ser. A 30 (1980/81), 461–468.Google Scholar
Elliott, P. D. T. A., Moreno, C. J. and Shahidi, F., On the absolute value of Ramanujan’s τ-function, Math. Ann. 266 (1984), 507–511.CrossRefGoogle Scholar
Gelbart, S. and Jacquet, H., A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Éc. Norm. Supér. (4) 11 (1978), 471–542.CrossRefGoogle Scholar
Goldfeld, D., Automorphic forms and L-functions for the group GL(n, R), Cambridge Studies in Advanced Mathematics, vol. 99 (Cambridge University Press, Cambridge, 2006).Google Scholar
Goldfeld, D. and Li, X., Voronoi formulas on GL(n), Int. Math. Res. Not. IMRN (2006), 1–25; Art. ID. 86295.Google Scholar
Green, B. and Tao, T., The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175 (2012), 541–566.Google Scholar
Hoffstein, J. and Ramakrishnan, D., Siegel zeros and cusp forms, Int. Math. Res. Not. IMRN 6 (1995), 279–308.Google Scholar
Holowinsky, R., A sieve method for shifted convolution sums, Duke Math. J. 146 (2009), 401–448.Google Scholar
Iwaniec, H., Introduction to the spectral theory of automorphic forms, Rev. Mat. Iberoam. (1995).Google Scholar
Iwaniec, H., Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17 (American Mathematical Society, Providence, RI, 1997).Google Scholar
Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Kim, H., Functoriality for the exterior square of GL4and symmetric fourth of GL2. With Appendix 1 by D. Ramakrishnan, and Appendix 2 by H. Kim and P. Sarnak, J. Amer. Math. Soc. 16 (2003), 139–183.CrossRefGoogle Scholar
Kim, H. and Shahidi, F., Functorial products for GL2×GL3and the symmetric cube for GL2. With an appendix by C.J. Bushnell and G. Henniart, Ann. of Math. (2) 155 (2002), 837–893.Google Scholar
Kim, H. and Shahidi, F., Cuspidality of symmetric power with applications, Duke Math. J. 112 (2002), 177–197.Google Scholar
Lau, Y. K. and Lü, G. S., Sums of Fourier coefficients of cusp forms, Q. J. Math. 62 (2011), 687–716.Google Scholar
Li, W., Newforms and functional equations, Math. Ann. 212 (1975), 285–315.Google Scholar
Liu, J., Wang, Y. and Ye, Y., A proof of Selberg’s orthogonality for automorphic L-functions, Manuscripta Math. 118 (2005), 135–149.CrossRefGoogle Scholar
Liu, J. and Ye, Y., Selberg’s orthogonality conjecture for automorphic L-functions, Amer. J. Math. 127 (2005), 837–849.Google Scholar
Liu, J. and Ye, Y., Perron’s formula and the prime number theorem for automorphic L-functions, Pure Appl. Math. Q. 3 (2007), 481–497; Special Issue: In honor of Leon Simon. Part 1.Google Scholar
Miller, S., Cancellation in additively twisted sums on GL(n), Amer. J. Math. 128 (2006), 699–729.Google Scholar
Miller, S. D. and Schmid, W., Summation formulas, from Poisson and Voronoi to the present, in Noncommutative harmonic analysis, Progress in Mathematics, vol. 220 (Birkhäuser, Boston, MA, 2004), 419–440.Google Scholar
Miller, S. D. and Schmid, W., Automorphic distributions, L-functions, and Voronoi summation for GL(3), Ann. of Math. (2) 164 (2006), 423–488.Google Scholar
Montgomery, H. L. and Vaughan, R. C., Multiplicative number theory I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97 (Cambridge University Press, Cambridge, 2007).Google Scholar
Murty, M. R., Oscillations of Fourier coefficients of modular forms, Math. Ann. 262 (1985), 431–446.Google Scholar
Murty, M. R. and Sankaranarayanan, A., Averages of exponential twists of the Liouville functions, Forum Math. 14 (2002), 273–291.Google Scholar
Perelli, A., On the prime number theorem for the coefficients of certain modular forms, in Elementary and analytic theory of numbers, Banach Center Publications, vol. 17 (PWN, Warsaw, 1982), 405–410.Google Scholar
Rankin, R. A., Contributions to the theory of Ramanujan’s function τ (n) and similar arithmetical functions. I. The zeros of the function ∑n=1∞τ (n)∕n s on the line ℜs = 13∕2. II. The order of the Fourier coefficients of integral modular forms, Math. Proc. Cambridge Philos. Soc. 35 (1939), 357–372.Google Scholar
Rankin, R. A., An Ω result for coefficients of cusp forms, Math. Ann. 103 (1973), 239–250.Google Scholar
Rankin, R. A., Sum of powers of cusp form coefficients. II, Math. Ann. 272 (1985), 593–600.CrossRefGoogle Scholar
Sarnak, P., Three lectures on the Möbius function randomness and dynamics. Available at www.math.ias.edu/files/wam/2011/PSMobius.pdf.Google Scholar
Sarnak, P. and Ubis, A., The horocycle flow at prime times. Preprint (2011),arXiv:1110.0777 [math.NT].Google Scholar
Selberg, A., Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47–50.Google Scholar
Soundararajan, K., Quantum unique ergodicity for SL2(ℤ)∖ℍ, Ann. of Math. (2) 172 (2010), 1529–1538.Google Scholar
Titchmarsh, E. C., The Theory of the Riemann zeta-function, second edition, (Clarendon Press, Oxford, 1986); revised by D. R. Heath-Brown.Google Scholar
Wu, J., Power sums of Hecke eigenvalues and application, Acta Arith. 137 (2009), 333–344.Google Scholar
Wu, J. and Xu, Z., Power sums of Hecke eigenvalues of Maass cusp forms, Ramanujan J., to appear.Google Scholar