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 $\lambda _{\pi _i}(n)$.