Consider \[ A(x) = \sum_{n\leq X} a(n) \quad\text{and}\quad \mathcal{A}(s) = \sum_{n=1}^\infty \frac{a(n)}{n^s}, \] a degree $d$ $L$-function (in the Selberg class). Friedlander-Iwaniec (https://doi.org/10.4153/CJM-2005-021-5) have showed that $A(X) = R(X) + \Delta(X)$, where \[ R(X) = X \ Res_{s=1} \frac{\mathcal{A}(s)}{s} \quad\text{and}\quad \Delta(X) = O(X^{\frac{d-1}{d+1}+\epsilon}). \]

We will focus mainly on the condition when $a(n) = \lambda_f (n)\lambda_g(n)$ where $\lambda_f(n)$ and $\lambda_g(n)$ are normalized Fourier coefficients of Hecke cusp forms (Maass orholomorphic). For this case we have \[ A(X) = C_f \mathbb{I}_{f=\overline{g}}X+O(X^{3/5+\epsilon}), \] where for $f = \overline{g}$, we have $C_f = \frac{L(1,sym^2 f)}{\zeta(2)}$ and $\mathbb{I}$ is the indicator function. Huang (Math. Ann. https://doi.org/10.1007/s00208-021-02186-7) considered the problem only when $f = \overline{g}$ and showed that \[ \Delta(X) = O(X^{3/5-1/560+\epsilon}) \]

In an upcoming work, with Kummari Mallesham, Ritabrata Munshi and Saurabh Kumar Singh, we will address this problem when $f \neq \overline{g}$.