Asymptotics of D(q)-pairs and triples via L-functions of Dirichlet characters: Asymptotics of D(q)-pairs and triples via L-functions

Let q be a non-zero integer. A D ( q )- m -tuple is a set of m distinct positive integers { a 1 , a 2 , ⋯ , a m } such that a i a j + q is a perfect square for all 1 ⩽ i < j ⩽ m . By counting integer solutions x ∈ [ 1 , b ] of congruences x 2 ≡ q ( mod b ) with b ⩽ N , we count D ( q )-pairs with both elements up to N , and give estimates on asymptotic behaviour. We show that for prime q , the number of such D ( q )-pairs and D ( q )-triples grows linearly with N . Up to a factor of 2, the slope of this linear function is the quotient of the value of the L -function of an appropriate Dirichlet character (usually a Kronecker symbol) and of ζ ( 2 ) .