Bound for $GL(3)$ symmetric square

Question

Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if

$$\sum_{n>0} \frac{|a_n|}{n^s}$$

and

$$\sum_{n>0} \frac{\left( \sum_{k \mid n} |a_k|\right)^2}{n^s}$$

converge for $\Re(s)>1$? or even for $\Re(s)>1+\delta$ for a quite small $\delta$?

This essentially amounts to say that the coefficients are constant on average. It seems to be known for Gelbart-Jacquet lifts by Rankin-Selberg properties, but is it known or at least expected for non-Gelbart-Jacquet lifts?

Answer 1

Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_m$ with unitary central character. In work of Takeda, it is shown that the (unramified part of the) $L$-function $L(s,\pi,\mathrm{Sym}^2)$ is holomorphic in the region $\mathrm{Re}(s) > 1-\frac{1}{2m}$. Taking $m=3$, we find that your first series converges for $\mathrm{Re}(s)>1$.

The second series is obviously a little trickier, but we can get there in the special case where $\pi$ is self-dual. In this case, $\pi$ is a Hecke character twist of the symmetric square lift of a cuspidal automorphic representation $\pi'$ on $\mathrm{GL}_2$. Suppose (for now) that $\pi=\mathrm{Sym}^2\pi'$ and $\pi$ has level 1 (so $\pi'$ is nondihedral). Then $\mathrm{Sym}^2\pi = 1\boxplus \mathrm{Sym}^4 \pi'$, in which case

$L(s,\pi,\mathrm{Sym}^2) = \zeta_F(s) L(s,\mathrm{Sym}^4\pi')$.

Note that $\mathrm{Sym}^4\pi'$ is a cuspidal automorphic representation of $\mathrm{GL}_5$. Thus $\Pi = 1\boxplus \mathrm{Sym}^4 \pi'$ is an automorphic representation of $\mathrm{GL}_6$, and the $L$-function

$L(s,\Pi\times\tilde{\Pi}) = \zeta_F(s)L(s,\mathrm{Sym}^4\pi')^2 L(s,\mathrm{Sym}^4\pi'\times \mathrm{Sym}^4\pi')$

converges absolutely for $\mathrm{Re}(s)>1$. The $n$-th Dirichlet coefficient $\lambda_{\Pi\times\tilde{\Pi}}(n)$ is bounded below by $|a_n|^2$. A Dirichlet convolution calculation shows that if $s>1$, then the series is bounded above by

$L(s,\Pi\times\tilde{\Pi})\zeta_F(s)^2$

(or maybe 2 needs to be replaced with a higher power?), which converges absolutely for $\mathrm{Re}(s)>1$.

Takeda, Shuichiro, The twisted symmetric square (L)-function of (\mathrm{GL}(r)), Duke Math. J. 163, No. 1, 175-266 (2014). ZBL1316.11037.