Dirichlet series, asymptotics, and statistics based on functoriality from GL(2)


Let \(\pi _i\), \(i=1,2,3\), be unitary automorphic cuspidal representations of \(GL_2({\mathbb {Q}}_{\mathbb {A}})\) with Fourier coefficients \(\lambda _{\pi _i}(n)\). Consider an automorphic representation \(\Pi \) which is equivalent to \(\wedge ^2(\mathrm{Sym}^3\pi _1)\), \(\pi _1\boxtimes \pi _2\), \(\pi _1\boxtimes \mathrm{Sym}^2\pi _2\), \(\wedge ^2(\pi _1\boxtimes \pi _2)\), or \(\pi _1\times \pi _2\times \pi _3\). Since the Dirichlet series of \(L(s,\Pi \times \widetilde{\Pi })\) is known to be complicated, a simpler Dirichlet series \(\sum \lambda (n)n^{-s}\) is defined and analytically continued in each case, which is closely related to \(L(s,\Pi \times \widetilde{\Pi })\) and catches the essence of the underlying functoriality. Asymptotics of \(\sum _{n\le x}\lambda (n)\) are proved. As applications, certain means, variance, and covariances of \(|\lambda _{\pi _i}(n)|^{k}\) for \(k=2,4,6\) and \(|\lambda _{\pi _i}(n^j)|^2\) for \(j=2,3,4\) are computed. These statistics provide a deep insight of the distribution of the GL(2) Fourier coefficients \(\lambda _{\pi _i}(n)\).

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  1. 1.

    The first author is partially supported by Natural Science Foundation of Shandong Province (Grant No. ZR2018MA003).


  1. 1.

    Asgari, M., and Raghuram, A.: A cuspidality criterion for the exterior square transfer of cusp forms on \(GL(4)\). In: J. Arthur, J.W. Cogdell, S. Gelbart, D. Goldberg, D. Ramakrishnan, J.-K. Yu, eds., On certain L-functions. Papers from the Conference in Honor of Freydoon Shahidi Held at Purdue University, West Lafayette, IN, July 23–27, 2007. Clay Math. Proc. 13, American Mathematical Society, 2011, Providence, pp. 33–53

  2. 2.

    Bombieri, E., Iwaniec, H.: On the order of \(\zeta \Big (\frac{1}{2} +i t \Big )\). Ann. Scuola. Norm. Super. Pisa Cl. Sci. 13(3), 449–472 (1986)

    MATH  Google Scholar 

  3. 3.

    Bombieri, E., Iwaniec, H.: Some mean-value theorems for exponential sums. Ann. Scuola. Norm. Super. Pisa Cl. Sci. 13(3), 473–486 (1986)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bourgain, J.: Decoupling, exponential sums and the Riemann zeta function. J. Am. Math. Soc. 30(1), 205–224 (2017)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Garrett, P.B.: Decomposition of Eisenstein series: Rankin triple products. Ann. Math. 125(2), 209–235 (1987)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Goldfeld, D., Hundley, J.: Automorphic Representations and \(L\)-Functions for the General Linear Group, vol. 1. Cambridge University Press, Cambridge (2011)

    Google Scholar 

  7. 7.

    Henniart, G.: Sur la fonctorialité, pour \(GL(4)\), donnée par le carré extérieur. Moscow Math. J. 9(1), 33–45 (2009)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Huxley, M.N.: Exponential sums and the Riemann zeta function. Proc. Lond. Math. Soc. 90(1), 1–41 (2005)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Ikeda, T.: On the functional equations of the triple \(L\)-functions. J. Math. Kyoto Univ. 29(2), 175–219 (1989)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Jiang, Y.J., Lü, G.S.: Sums of coefficients of \(L\)-functions and applications. J. Number Theory 171, 56–70 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kim, H.: Functoriality for the exterior square of \(GL_4\) and symmetric fourth of \(GL_2\). J. Am. Math. Soc. 16, 139–183 (2003)

    Article  Google Scholar 

  12. 12.

    Kim, H., Shahidi, F.: Functorial products for \(GL_2 \times GL_3\) and the symmetric cube for \(GL_2\) (with an appendix by C.J. Bushnell and G. Henniart). Ann. Math. 155, 837–893 (2002)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Landau, E.: Über die Anzahl der Gitterpunkte in gewisser Bereichen (Zweite Abhandlung). Ges. d. Wiss Göttingen Math. Phys. Klasse. 1915, 209–243 (1915)

    MATH  Google Scholar 

  14. 14.

    Lao, H.X., Sankaranarayanan, A.: The average behavior of Fourier coefficients of cusp forms over sparse sequences. Proc. Am. Math. Soc. 137, 2557–2565 (2009)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lao, H.X., Sankaranarayanan, A.: The distribution of Fourier coefficients of cusp forms over sparse sequences. Acta Arith. 163(2), 101–109 (2014)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Lao, H.X., McKee, M., Ye, Y.B.: Asymptotics for cuspidal representations by functoriality from GL(2). J. Number Theory 164, 323–342 (2016)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Lau, Y.-K., Lü, G.S.: Sums of Fourier coefficients of cusp forms. Q. J. Math. 62, 687–716 (2011)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Lau, Y.-K., Lü, G.S., Wu, J.: Integral power sums of Hecke eigenvalues. Acta Arith. 150(2), 193–207 (2011)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Li, X.Q.: Bounds for \(GL(3)\times GL(2)\)\(L\)-functions and \(GL(3)\)\(L\)-functions. Ann. Math. 173, 301–336 (2011)

    MathSciNet  Article  Google Scholar 

  20. 20.

    McKee, M., Sun, H.W., Ye, Y.B.: Improved subconvexity bounds for \(GL(2) \times GL(3)\) and \(GL(3)\) L-functions by weighted stationary phase. Trans. Am. Math. Soc. 370(5), 3745–3769 (2018)

    Article  Google Scholar 

  21. 21.

    Munshi, R.: The circle method and bounds for \(L\)-functions-III: \(t\)-aspect subconvexity for \(GL(3)\)\(L\)-functions. J. Am. Math. Soc. 28, 913–938 (2015)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Nunes, R.M.: On the subconvexity estimate for self-dual \(GL(3)\)\(L\)-functions in the \(t\)-aspect. arXiv:1703.04424

  23. 23.

    Perelli, A.: General \(L\)-functions. Ann. Mat. Pur. Appl. 130(4), 287–306 (1982)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Piatetski-Shapiro, I., Rallis, S.: Rankin triple \(L\) functions. Comput. Math. 64(1), 31–115 (1987)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Rankin, R.A.: Contributions to the theory of Ramanujan’s function \(\tau (n)\) and similar arithmetical functions II. The order of the Fourier coefficients of the integral modular forms. Math. Proc. Camb. Philos. Soc. 35, 357–372 (1939)

    Article  Google Scholar 

  26. 26.

    Rudnick, Z., Sarnak, P.: Zeros of principal \(L\)-functions and random matrix theory. Duke Math. J. 81, 269–322 (1996)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Selberg, A.: Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist. Arch. Math. Naturvid. 43, 47–50 (1940)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Sun, H.W., Ye, Y.B.: Further improvement on bounds for \(L\)-functions related to \(GL(3)\). Int. J. Number Theory 15(7), 1487–1517 (2019)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Watson, T.C.: Rankin triple products and quantum chaos. Princeton University, Thesis (2002). arXiv:0810.0425

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Correspondence to Yangbo Ye.

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Lao, H., Ye, Y. Dirichlet series, asymptotics, and statistics based on functoriality from GL(2). Res. number theory 6, 37 (2020). https://doi.org/10.1007/s40993-020-00212-2

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  • Automorphic representation
  • Cuspidal representation
  • Functoriality
  • Dirichlet series
  • GL(2)
  • Fourier coefficient
  • Asymptotic expansion
  • Statistics

Mathematics Subject Classification

  • 11F70
  • 11F66
  • 11F30