Major arcs and moments of arithmetical sequences

TL;DR

This paper develops new estimates for the first two moments of arithmetical sequences in progressions using a generalized Vaughan's major arcs approximation, with applications to the divisor function sequence.

Contribution

It introduces a generalized Vaughan's major arcs method for moment estimates, extending previous approaches to broader classes of sequences and moduli.

Findings

Unconditional results for the sequence τ_k(n) in various moduli

Improved moment estimates using generalized major arcs

Application to norm form related sequences

Abstract

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.