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Let Q (x) be a positive definite integral quadratic form with the determinant D being squarefree, and r (n , Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function d (n) and Fourier coefficients r (n , Q). With more efforts, our method should have a number of applications for other multiplicative functions. [ABSTRACT FROM AUTHOR]

Let λ f (n) be the normalized Fourier coefficients of a Hecke-Maass or Hecke holomorphic cusp form f for congruence group Γ 0 (N) with level N and nebentypus χ N. Let Q (x) be a positive definite integral quadratic form, and r (n , Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply Jutila's circle method to derive a sharp bound for the shifted convolution sum of Fourier coefficients λ f (n) and r (n , Q). We generalize and improve previous results without the Ramanujan conjecture. [ABSTRACT FROM AUTHOR]

Let d k (n) denote the k -th divisor function. In this paper, we study the asymptotic formula of the sum ∑ 1 ≤ n 1 , n 2 , ... , n l ≤ x d k (n 1 2 + n 2 2 + ⋯ + n l 2) , where n 1 , n 2 , ... , n l ∈ Z + , k ≥ 4 and l ≥ 3 are integers. Previously only the cases of k = 2 , 3 are studied. [ABSTRACT FROM AUTHOR]