T(h,N) equals (16/ζ(2)) N² times the sum of 1/d over divisors d of h, plus an error O_ε(N^ε (N + h)).
Estermann,On the Representations of a Number as the Sum of Two Products, Proc
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Counting $2 \times 2$ integer matrices with a given determinant
T(h,N) equals (16/ζ(2)) N² times the sum of 1/d over divisors d of h, plus an error O_ε(N^ε (N + h)).