The variance asymptotic conjecture for sums of two squares

Let $b(n)$ be the indicator of integers representable as sums of two squares. Let $B(x)=\sum_{n\leq x}b(n)$, let $I(x;H)$ be the paper's smooth approximation to $B(x+H)-B(x)$, and define

$$V_b(X;H):=\frac{1}{X}\int_X^{2X}\big(B(x+H)-B(x)-I(x;H)\big)^2\,dx.$$

Let $K$ be the Landau–Ramanujan constant and let $G$ denote the function defined in the paper. The variance asymptotic conjecture. For fixed $\delta\in(0,1)$, as $X\to\infty$ with $H=X^\delta$,

$$V_b(X;H)=\left(KG(1-\delta)+o(1)\right)\frac{H}{\sqrt{\log X}}.$$

The theorem motivating this conjecture establishes the corresponding function-field limiting structure, while the integer asymptotic remains open.