Variance conjecture for divisor sums in arithmetic progressions

Let $k\geq 2$ be fixed, and let

$$d_k(n):=\#\{(a_1,\ldots,a_k)\in\mathbb{N}^k:a_1\cdots a_k=n\}$$

be the $k$-fold divisor function. For integers $q\geq 1$ and $X\geq 1$, define

$$v_k(q;X):=\sum_{\substack{1\leq a\leq q\\(a,q)=1}}\left|\sum_{\substack{n\equiv a\pmod q\ \leq X}}d_k(n)-\frac{1}{\phi(q)}\sum_{\substack{(n,q)=1\ \leq X}}d_k(n)\right|^2.$$

Also define

$$a_k(q):=\lim_{s\to1^+}(s-1)^{k^2}\sum_{\substack{n\geq1\\(n,q)=1}}\frac{d_k(n)^2}{n^s},$$

where $\gamma_k(c)$ is a piecewise polynomial of degree $k^2-1$.

Variance conjecture for divisor sums in arithmetic progressions. If $X,q\to\infty$ in such a way that $\frac{\log X}{\log q}\to c\in(0,k)$, then

$$v_k(q;X)\sim a_k(q)\gamma_k(c)X(\log q)^{k^2-1}.$$

The function $\gamma_k(c)$ is positive for $c\in(0,k)$ and is described more fully later in the source. This conjecture predicts the asymptotic variance of $k$-fold divisor sums over reduced residue classes when both the modulus and the summation range grow. The range $c<1$ corresponds to progressions containing at most one term, while values of $c$ just above $1$ concern moduli close to, but smaller than, $X$. The paper proves an averaged version in a restricted range; the full pointwise asymptotic stated here remains open.