On the error term of a lattice counting problem

On the error term of a lattice counting problem, II

“…This work is the continuation of the paper [3] in which the following problem is studied. For integer T 1, we let F (T ) := {a/b : (a, b) ∈ Z 2 , 0 a < b T, (a, b) = 1} be the set of Farey fractions.…”

“…where C a,b (T ) := F (T ) ∩ 1 − a 2 /b 2 , 1 . As it is mentioned in [3], this quantity C(T ) appears naturally in some counting problems for twodimensional lattices and the main term of the asymptotic formula for C(T ) can be expressed via the cardinality More precisely, it is shown [3, Theorem 1.1] that, unconditionally…”

“…For integer T 1, we let F (T ) := {a/b : (a, b) ∈ Z 2 , 0 a < b T, (a, b) = 1} be the set of Farey fractions. We also define As it is mentioned in [3], this quantity C(T ) appears naturally in some counting problems for twodimensional lattices and the main term of the asymptotic formula for C(T ) can be expressed via the cardinality F (T ) := #F (T ) of the set of Farey fractions and also second moment of the Farey fractions in [0, 1 2 ]:…”

“…where ρ(t) := exp (log t) 1/2 (log log t) 5/2+o (1) and the authors derived in [3,Theorem 1.4] the estimate…”

Sums of certain fractional parts

“…where P (x) is a polynomial of degree 2 or 3 with small positive leading coefficient. The method was then generalized in [3] to any polynomial of degree 2, and we use here the Weyl's schift to extend the results to smooth functions. Unfortunately, as often in exponential sums estimates, the secondary terms remain too weak to be really efficients in practice.…”