On Differences of Semicubical Powers

Abstract.

For X,Y,Δ>0, let and define I ₈(X,Y,Δ) to be the cardinality of the set. In this paper it is shown that, for ε>0, Y ²/X ³ = O(Δ), Δ = O(Y ³/X ³) and X = O (Y ²), one has I ₈(X,Y,Δ) = O(X ² Y ² + X ^ε min (X ^{3/2} Y ³, Δ X ^{11/2} Y ^{−1}) + X ^ε min (Δ^{1/3} X ² Y ³, Δ X ^{14/3} Y ^{1/3})), with the implicit constant depending only on ε. There is a brief report on an application of this that leads, by way of the Bombieri-Iwaniec method for exponential sums, to some improvement of results on the mean squared modulus of a Dirichlet L-function along a ‘short’ interval of its critical line.