Major arcs and moments of arithmetical sequences

We give estimates for the first two moments of arithmetical sequences in progressions. Instead of using the standard approximation, we work with a generalization of Vaughan's major arcs approximation which is similar to that appearing in earlier work of Browning and Heath-Brown on norm forms. We apply our results to the sequence $\tau_k(n)$, and obtain unconditional results in a wide range of moduli.