On Averages of Shifted Convolutions with Applications to $GL(2)$ and $GL(3)$ Fourier Coefficients

In this paper, we study the average of shifted sum for general multiplicative functions. As applications, we prove non-trivial upper bounds for weighted averages of shifted convolutions involving $GL(2)$ and $GL(3)$ Fourier coefficients without smoothing. We apply square-root cancellation on average over short intervals for $GL(2)$ Fourier coefficients with the standard Hardy-Littlewood circle method.