Jiseong Kim, University at Buffalo, The State University of New York

Let $f$ be a holomorphic Hecke cusp form or Hecke-Maass cusp form for $SL(2,\mathbb{Z}),$ and let $\lambda_{f}(n)$ be the $n$th Hecke eigenvalue. Rankin and Selberg showed that
$$\sum_{1 \leq m \leq X} \lambda_f(m)^2=\frac{\textrm{Res}_{s=1} L\left(\textrm{sym}^2 f, s\right)}{\zeta(2)} X+O_f\left(X^{3 / 5}\right)$$
where $L\left(\textrm{sym}^2 f, s\right)$ is the symmetric square $L$-function of $f, \zeta(s)$ is the Riemann zeta function, and recently, Huang improved the exponent $3/5$ to $3/5-1/560+o(1).$ The Rankin-Selberg problem is to improve the exponent of the error term. In this talk, we discuss the Rankin-Selberg problems in families by applying the trace formulae and the Lindel\"of-on-average bound.