In this talk, we shall introduce our recent work concerning cancellation inadditively twisted sums on $\mathrm{GL}(m)\times\mathrm{GL}(m^{\prime})$. We describe a new method to obtain upper bounds for exponential sums with multiplicative coefficients without the Ramanujan conjecture. We verify these hypothesis for (with mild restrictions) the Rankin-Selberg $L$-functions attached to two cuspidal automorphic representations.