Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T08:53:59.373Z Has data issue: false hasContentIssue false
  • English
  • Français

Cancellation of two classes of dirichlet coefficients over Beatty sequences

Part of: Sequences and sets Discontinuous groups and automorphic forms Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  20 April 2021

Qiang Ma
Qiang Ma*
Affiliation:
School of Mathematics, Shandong University, Jinan250100, China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $\pi $ be an automorphic irreducible cuspidal representation of $\mathrm{GL}_{m}$ over $\mathbb {Q}$ . Denoted by $\lambda _{\pi }(n)$ the nth coefficient in the Dirichlet series expansion of $L(s,\pi )$ associated with $\pi $ . Let $\pi _{1}$ be an automorphic irreducible cuspidal representation of $\mathrm{SL}(2,\mathbb {Z})$ . Denoted by $\lambda _{\pi _{1}\times \pi _{1}}(n)$ the nth coefficient in the Dirichlet series expansion of $L(s,\pi _{1}\times \pi _{1})$ associated with $\pi _{1}\times \pi _{1}$ . In this paper, we study the cancellations of $\lambda _{\pi }(n)$ and $\lambda _{\pi _{1}\times \pi _{1}}(n)$ over Beatty sequences.

Keywords

Automorphic representationDirichlet coefficientsBeatty sequences

MSC classification

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11B83: Special sequences and polynomials
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 234 - 252
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abercrombie, A. G., Banks, W. D., and Shparlinski, I. E., Arithmetic functions on Beatty sequences. Acta Arith. 136(2009), 81–89.CrossRefGoogle Scholar
Banks, W. D. and Shparlinski, I. E., Character sums with Beatty sequences on burgess-type intervals. Cambridge University Press, Cambridge, MA, 2009, 15–21.Google Scholar
Barthel, L. and Ramakrishnan, D., A nonvanishing result for twists of $L$ -functions of $GL(n)$ . Duke Math. J. 74(1994), 681–700.CrossRefGoogle Scholar
Besicovitch, A. S., Sets of fractional dimensions. IV: on rational approximation to real numbers. J. Lond. Math. Soc. 9(1934), 126–131.CrossRefGoogle Scholar
Bump, D., Automorphic forms and representations. Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, Cambridge, UK, 1997.CrossRefGoogle Scholar
Davenport, H., On certain exponential sums. J. Reine Angew. Math. 169(1933), 158–176.CrossRefGoogle Scholar
Deligne, P., La conjecture de Weil I. Inst. Hautes tudes Sci. Publ. Math. 48(1974), 273–308.CrossRefGoogle Scholar
Friedlander, J. and Iwaniec, H., Opera de cribro. American Mathematical Society Colloquium Publications, 57, American Mathematical Society, Providence, RI, 2010.Google Scholar
Gelbart, S. and Jacquet, H., A relation between automorphic representations of $GL(2)$ and $GL(3)$ . Ann. Sci. École Norm. Sup. 11(1978), 471–552.10.24033/asens.1355CrossRefGoogle Scholar
Güloǧlu, A. M. and Nevans, C. W., Sums of multiplicative functions over a Beatty sequence. Bull. Aust. Math. Soc. 78(2008), 327–334.CrossRefGoogle Scholar
Hall, R. R. and Tenenbaum, G., Divisors. Cambridge Tracts in Mathematics, 90, Cambridge University Press, Cambridge, MA, 1988.CrossRefGoogle Scholar
Hecke, E., Theorie der Eisensteinsche Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik. Abh. Math. Abh. Math. Semin. Univ. Hamb. 5(1927), 199–224.CrossRefGoogle Scholar
Iwaniec, H., Topics in classical automorphic forms. Graduate Studies in Mathematics, 17, American Mathematical Society, Providence, RI, 1997.Google Scholar
Iwaniec, H., Spectral methods of automorphic forms. Graduate Studies in Mathematics, 53, American Mathematical Society, Providence, RI, 2002.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic number theory. American Mathematical Society Colloquium Publications 53, American Mathematical Society, Providence, RI, 2004.Google Scholar
Jacquet, H. and Shalika, J. A., On Euler products and the classification of automorphic representations I. Amer. J. Math. 103(1981), 499–558.CrossRefGoogle Scholar
Jarnŕk, V., Diophantische Approximationen und Hausdorffsches Mass. Mat. Sb. 36(1929), 371–382.Google Scholar
Jiang, Y. J. and Lü, G. S., The generalized Bourgain-Sarnak-Ziegler criterion and its application to additively twisted sums on $GL(m)$ . Sci. China Math. 63(2020). https://doi.org/10.1007/s11425-020-1717-1.Google Scholar
Khintchine, A., Einige Sätzeüber Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math. Ann. 92(1924), 115–125.CrossRefGoogle Scholar
Kim, H. H., Functoriality for the exterior square of $GL(4)$ and the symmetric fourth of $GL(2)$ . J. Amer. Math. Soc. 16(2003), 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak.CrossRefGoogle Scholar
Kim, H. H., A note on Fourier coefficients of cusp forms on $GL(n)$ . Forum Math. 18(2006), 115–119.CrossRefGoogle Scholar
Kim, H. H. and Shahidi, F., Cuspidality of symmetric powers with applications. Duke Math. J. 112(2004), 177–197.Google Scholar
Kloosterman, H. D., Asymptotische Formeln für die Fourierkoeffizienten ganzer Modulformen. Abh. Math. Abh. Math. Semin. Univ. Hamb. 5(1927), 337–352.CrossRefGoogle Scholar
Kurpers, L. and Niederreiter, H., Uniform distribution of sequences. Pure and Applied Mathematics, Wiley-Interscience, New York, NY, 1974.Google Scholar
Lao, H. X., Oscillations of coefficients of primitive cusp form over some special sequences. Ramanujan J. 19(2009), 339–350.CrossRefGoogle Scholar
Lü, G. S., On sums involving coefficients of automorphic $L$ -functions. Proc. Amer. Math. Soc. 137(2009), 2879–2887.CrossRefGoogle Scholar
Lü, G. S., Estimates for coefficients of certain $\ L$ -functions. Monatsh. Math. 181(2016), 657–674.CrossRefGoogle Scholar
Lü, G. S., Exponential sums with Fourier coefficients of automorphic forms. Math. Z. 289(2018), 267–278.CrossRefGoogle Scholar
Luo, W. Z., Rudnick, Z., and Sarnak, P., On Selberg’s eigenvalue conjecture. Geom. Funct. Anal. 5(1995), 387–401.CrossRefGoogle Scholar
Miller, S., Cancellation in additively twisted sums on $GL(n)$ . Amer. J. Math. 128(2006), 699–729.CrossRefGoogle Scholar
Roth, K. F., Rational approximations to algebraic numbers. Mathematika 2(1955), 1–20.CrossRefGoogle Scholar
Rudnick, Z. and Sarnak, P., Zeros of principal $L$ -functions and random matrix theory. Duke Math. J. 81(1996), 269–322.10.1215/S0012-7094-96-08115-6CrossRefGoogle Scholar
Sun, Q. F., Cancellation of cusp forms coefficients over Beatty sequences on $GL(m)$ . Canad. Math. Bull. 54(2011), 757–762.CrossRefGoogle Scholar
Technau, M. and Zafeiropoulos, A., Metric results on summatory arithmetic functions on beatty sets. Acta Arith. 197(2021), 93–104.CrossRefGoogle Scholar
Vinogradov, I. M., The method of trigonometrical sums in the theory of numbers. Dover Publications, Mineola, NY, 2004.Google Scholar
Weil, A., On some exponential sums. Proc. Natl. Acad. Sci. 34(1948), 204–207.CrossRefGoogle ScholarPubMed