On the error term of a lattice counting problem, II

TL;DR

This paper improves the error term in a lattice counting problem under the Riemann Hypothesis by employing Weyl's bound and Popov's device, leading to more precise asymptotic estimates.

Contribution

It introduces a novel combination of Weyl's bound and Popov's method to refine the error term in lattice counting asymptotics under RH.

Findings

Enhanced error bounds under RH

Application of Weyl's bound to polynomial exponential sums

Use of Popov's device for fractional parts of polynomials

Abstract

Under the Riemann Hypothesis, we improve the error term in the asymptotic formula related to the counting lattice problem studied in a first part of this work. The improvement comes from the use of Weyl's bound for exponential sums of polynomials and a device due to Popov allowing us to get an improved main term in the sums of certain fractional parts of polynomials.