On the error term of a lattice counting problem, II

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.2010 Mathematics Subject Classification. Primary 11L07, 11L15, 11P21. Key words and phrases. Lattices, Weyl's bound for exponential sums, sums of fractional parts of polynomials.