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Description

A shifted convolution problem seeks an asymptotic formula for sums involving the product of two arithmetic functions whose arguments differ by an additive shift. Such problems arise naturally in the study of correlations of arithmetic functions and are closely connected to moments of $L$-functions. In joint work with Valentin Blomer, we investigate the shifted convolution of the divisor function with Fourier coefficients of $GL(3)$ automorphic forms, resolving the final remaining case of shifted convolution problems for $GL(3)\times GL(2)$. The proof relies on two intertwined applications of different delta symbol methods. As an application, we obtain an asymptotic formula for central values of $L$-functions associated with a $GL(3)$ automorphic form twisted by Dirichlet characters of modulus $q\leq Q$.