The critical-window profile for $d_k$ in short intervals

Abstract

We establish an almost-all transition theorem for the $k$-fold divisor function in short intervals. Let $k\geq2$ be fixed, let $\ell=\log\log X$, and set \[ D_k(X)=(\log X)^{k\log k-k+1}, \qquad M_k(x)=\frac1x\sum_{x<n\leq2x}d_k(n). \] The critical scale is $D_k(X)$. In the bounded part of the transition window, put \[ A(X)=\frac1{\sqrt{\ell}}\log\frac{h}{D_k(X)} \] and assume that $A(X)=O(1)$. Then, for almost all integers $x\in[X,2X]$, \[ \frac1h\sum_{x<n\leq x+h}d_k(n) = \left(\Phi_{\rm G}\left(\frac{A(X)}{\sqrt{k}\log k}\right)+o(1)\right) M_k(x), \] where $\Phi_{\rm G}$ denotes the standard Gaussian distribution function. Above the window, namely when \[ \frac{\log(h/D_k(X))}{\sqrt{\ell}}\to+\infty, \] the full long average is recovered for almost all $x$.