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

Abstract

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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Notes

  1. 1.

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

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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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Keywords

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

Mathematics Subject Classification

  • 11F70
  • 11F66
  • 11F30