Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients.
Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know the lower bound $\sum_{m^2n\leq x}|A(m,n)|^2\gg_{\delta} x^{1-\delta}$
Is something known about the Dirichlet series (functional equation, meromorphicity etc.) $$\sum_{n=1}^{\infty}\frac{|A(1,n)|^2}{n^s}?$$
I know that the $GL(2)$ analogues for both questions are well known.
For $(1)$, I believe the size would still be $\gg x^{1-\epsilon}$. However, if you allow $$\sum_{p\sim x} |A(p,1)|^2+A|(p^2,1)|^2$$ a lower bound of similar sort is obtained by Blomer-Maga's paper Corollary $4.3$. (In any case, one elementary way to know the average size of the Hecke eigenvalues is by writing them in Satake parameters using Shntani's formula.)
For $(2)$, the Dirichlet series appears in the Rankin-Selberg convolution of $$\langle \mathbb{P}f,\overline{\mathbb{P}f}\rangle,$$ where $\mathbb{P}$ is the standard projection operator from $\mathrm{GL}_3$ to $\mathrm{GL}_1$ which comes in the definition of global zeta integral. For instance, see chapter $10$ of Goldfeld's automorphic form book. Thus one can obtain functional equation and meromorphic continuation from the above.
The prime number theorem for $L(s,f\times\bar{f})$ and the truth of Hypothesis H for $\mathrm{GL}_3$ automorphic forms yields the asymptotic
$\sum_{p\leq x}|A(1,p)|^2\sim \frac{x}{\log x}$.
(See https://reader.elsevier.com/reader/sd/pii/S0022314X14000675?token=95B6184DD5B464DBE2DE74CD9D5F10C468E3C28DD137E91760946605ADD19AE8A34F21C0B8FA5F46CBE2B5F58FA6E858.) It follows from the work in Brumley's PhD thesis (see https://arxiv.org/abs/math/0510089) that there exists an Euler product $H(s,f)$ such that for $\mathrm{Re}(s)\geq 8/9$, we have the factorization
$(*)\qquad\sum_{n=1}^{\infty}\frac{|A(1,n)|^2}{n^s} = H(s,f)L(s,f\times\bar{f})$,
where $H(1,f)>0$ and $|H(s,f)|\ll_{\epsilon}\lambda^{\epsilon}$ for $\mathrm{Re}(s)\geq 8/9$ (here, $\lambda$ is the Laplace eigenvalue of $f$). This factorization leads to the asymptotic
$\sum_{n\leq x}|A(1,n)|^2\sim x \cdot H(1,f) \cdot \mathop{\mathrm{Res}}_{s=1}L(s,f\times\bar{f}).$
One can find explicit bounds on the error term using standard contour integration techniques and the convexity bound for $L(s,f\times\bar{f})$. Moreover, the functional equation for $L(s,f\times\bar{f})$ induces a functional equation for $(*)$, albeit in a limited region. The factorization $(*)$ would hold in the larger region $\mathrm{Re}(s)>1/2$ if we knew the generalized Ramanujan conjecture for $f$.
