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In this paper, we derived that a sufficiently large integer N can always be represented as the sum of three generalized D. H. Lehmer numbers. As a consequence, we deduced Lu and Yi's original result (Monatsh. Math., 159 (2010), 45-58). [ABSTRACT FROM AUTHOR]

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]

In this paper, we show that every sufficiently large even integer can be represented as eight prime cubes and k 1 powers of 2, and every pair of sufficiently large even integers can be represented as a pair of eight prime cubes and k 2 powers of 2, where k 1 = 28 and k 2 = 98. These results improve previous results k 1 = 30 and k 2 = 335 , respectively. [ABSTRACT FROM AUTHOR]