ºÚÁÏÍø´óʼÇ

2:00pm, Wednesday 9 July

Abstract

Let △ denote the integers represented by the quadratic form \( x^2 +xy+y^2\) and \(□_2\) denote the numbers represented as a sum of two squares. For a non-zero integer a, let \(S(△, □_2, a)\) be the set of integers n such that n ∈ △, and n + a ∈ \(□_2\). We conduct a census of \(S(△, □_2, a)\) in short intervals by showing that there exists a constant \( H_a >  0\) with

\( \#S(△, □_2, a) ∩ [x, x + H_a · x^{5/6}· log^{19} x] ≥ x^{5/6−ε} \)

for large x. To derive this result and its generalization, we utilize a theorem of Tolev (2012) on sums of two squares in arithmetic progressions and analyse the behavior of a multiplicative function found in Blomer, Br¨udern & Dietmann (2009). Our work extends a classical result of Estermann (1932) and builds upon work of M¨uller (1989)