GAPS BETWEEN HEXAGONAL AND SQUARE LATTICES

Abstract Let $\triangle $ denote the integers represented by the quadratic form $x^2+3y^2$ and denote the numbers represented as a sum of two squares. For a nonzero integer a , let be the set of integers n such that $n \in \triangle $ and . We conduct a census of in short intervals by showing that there exists a constant $H_{a}> 0$ with for large x . To derive this result and its generalisation, we use a theorem of Tolev [‘On the remainder term in the circle problem in an arithmetic progression’, Proc. Steklov Inst. Math. 276 (2012), 261–274].