Just very quick, as I don't have time. The terms in Vaughan's identity are not random sums: they are convolutions of simpler and/or shorter range arithmetic functions. Convolution allows the separation of variables in the sum, and hence it allows a more efficient estimation of the sum. A classical and prime example is Dirichlet's hyperbola method.

The terms in Vaughan's identity are not random sums: they are convolutions of simpler and/or shorter range arithmetic functions. Convolution allows the separation of variables in the sum, and hence it allows a more efficient estimation of the sum. A classical and prime example is Dirichlet's hyperbola method.