The divisor-sum variance asymptotic conjecture

For $z>0$, let $d_z(n)$ be the generalized divisor function, define $D_z(x)=\sum_{n\leq x}d_z(n)$, and let $\overline D_z(x)$ be the contour approximation defined in the paper. Set

$$\Delta_z(x):=D_z(x)-\overline D_z(x),\qquad \Delta_z(x;H):=\Delta_z(x+H)-\Delta_z(x),$$

and

$$V_{d_z}(X;H):=\frac{1}{X}\int_X^{2X}\Delta_z(x;H)^2\,dx.$$

Write

$$a_z:=\lim_{s\to1^+}(s-1)^{z^2}\sum_{n=1}^{\infty}\frac{d_z(n)^2}{n^s},$$

and let $\alpha_1^{(z)}$ be the random variable used in the paper's limiting distribution. The divisor-sum variance asymptotic conjecture. For fixed $z>0$ and fixed $\delta\in(0,1)$, with $H=X^\delta$,

$$V_{d_z}(X;H)=a_z\left(\frac{\mathbb{P}(\alpha_1^{(z)}\leq1-\delta)}{\Gamma(z^2)}+o(1)\right)H(\log X)^{z^2-1}$$

as $X\to\infty$. This generalizes a conjecture previously made for integer $z$; the stated extension remains open.