Question about mean square estimate for sums of Dirichlet coefficients of Symmetric Power $L$-functions

Question

I have a question related to Coefficients of Symmetric power $L$-functions and I would be grateful if you could answer it. Let $\lambda_{Sym^rf}(n)$ be the $n$th Dirichlet coefficient of $L(Sym^rf,s).$ I want to get a bound for the following sum $\sum_{n \leq x}|\lambda_{Sym^rf}(n)|^2 .$ So, I started by the paper (SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS) of YUK-KAM LAU AND GUANGSHI LU and I found that $$|\lambda_{Sym^rf}(n)|^2 \leq \lambda_{Sym^rf \times Sym^rf}(n),$$ where $\lambda_{Sym^rf \times Sym^rf}(n)$ is the $n$th Dirichlet coefficients of the Rankin-Selberg $L$-functions attached to $Sym^rf$ and $Sym^rf$ and $f$ is a primitive cusp form for the full modular group $SL_2(Z).$ Also, I read the paper of Huixue Lao (Mean square estimates for coefficients of symmetric power $L$-functions) and I found that $$|\lambda_{Sym^rf \times Sym^rf}(n)|\leq d_{r+1}(n)^2,$$ where $d_{r+1}(n)$ is the $n$ th coefficient of the Dirichlet series $\zeta^{r+1}(s).$ My question is: Is there a bound for the sum $\sum_{n \leq x} d_{r+1}(n)^2$ or directly a bound for the sum $$\sum_{n \leq x}|\lambda_{Sym^rf}(n)|^2? $$

Answer 1

It is classical to bound $\sum_{n\leq x}d_{r+1}(n)^2$. Using the fact that $$\sum_n\frac{d_{r+1}(n)^2}{n^s}=\prod_p\left(1+\frac{(r+1)^2}{p^s}+\frac{O_r(1)}{p^{2s}}\right)$$ one can show by Mellin transform techniques (i.e. by the residue theorem) that $$ \sum_{n\leq x}d_{r+1}(n)^2 \sim c_r x(\log x)^{r^2+2r}, $$ where $c_r>0$ can be given explicitly (it is a scaled residue at $s=1$ of a certain Euler product). More precise asymptotic expansions can also be given.

So you can definitely bound $\sum_{n \leq x}|\lambda_{Sym^rf}(n)|^2$ as $$ \sum_{n \leq x}|\lambda_{Sym^rf}(n)|^2 \ll_r x(\log x)^{r^2+2r}. $$ However, this will be crude in general, because we expect the coefficients $\lambda_{Sym^rf}(n)$ to oscillate. More precisely, by the Langlands functoriality conjecture, we expect the corresponding $L$-function $\sum_n\lambda_{Sym^rf}(n)n^{-s}$ to be the $L$-function of an automorphic representation $\pi$, and in the generic case we expect $\pi$ to be cuspidal (there are exceptions of course). Assuming this is the case, including the cuspidality of $\pi$, then $L(s,\pi\otimes\tilde\pi)$ is analytic on $\mathbb{C}\setminus\{1\}$ with a simple pole at $s=1$, and its $n$-th Dirichlet coefficient is bounded from below by $|\lambda_{Sym^rf}(n)|^2$ when $n$ is supported on the unramified primes (cf. Proposition 6 in Molteni: Upper and lower bounds at $s=1$ for certain Dirichlet series with Euler product). It follows, using standard bounds for the local parameters at the ramified primes, that in this case (i.e. when $\pi$ exists and it is cuspidal) $$ \sum_{n \leq x}|\lambda_{Sym^rf}(n)|^2 \ll_{f,r} x. $$ The implied constant can be estimated as $\ll_{\epsilon,r} C^\epsilon$, where $C$ is the analytic conductor of the original $\mathrm{GL}_2$ cusp form $f$ (cf. Theorem 2 in Li: Upper bounds on $L$-functions at the edge of the critical strip). For $r$ small, one can weaken the hypotheses due to known cases of functoriality.

P.S. My answer is valid for any $\mathrm{GL}_2$ cusp form $f$, including Maass forms of arbitrary level and nebentypus.