Correlations of multiplicative functions with automorphic L-functions

Let ${\lambda }_{\varphi }(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke–Maass cusp form on ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$, and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a non-trivial upper bound for the correlation $\sum_{n \leq X}f(n)\lambda_{\phi}(n+h)$uniformlyin$0<|h|\ll X$. As applications, we consider some special cases, including $\lambda_{\pi}(n), ,\mu(n)\lambda_{\pi}(n)$andanydivisor-boundedmultiplicativefunction\mathrm{.}Here$\lambda_{\pi}(n)$denotesthe$n$-thDirichletcoefficientof$$\text{GL}$_m$automorphic$L$-function$L(s,\pi)$ for an automorphic irreducible cuspidal representation $\pi$, and $\mu (n)$ denotes the M"obius function. In particular, some savings are achieved for shifted convolution problems on ${\rm GL}_m\times {\rm GL}_2, (m\geq 4)$ and Hypothesis C for the first time.