I'm interested in summing $\tau(m)$, the number of positive divisors of $m$, not over all integers in an interval but rather over only the integers with the most divisors. More specifically:
Given a large positive integer $x$, let $n_1, n_2, \dots, n_x$ be the integers $1, 2, \dots, x$, arranged so that $\tau(n_1) \ge \tau(n_2) \ge \cdots \ge \tau(n_x)$. I'm interested in the behavior of the function $$ f(x,y) = \sum_{j=1}^y \tau(n_j). $$ For example, when $y=1$ then $f(x,1)$ is just the maximal order of the divisor function $\tau$; when $y=x$ we have $f(x,x) \sim x\log x$ by Dirichlet. I believe that we have $f(x,y) \sim x\log x$ even when $y$ is as small as $x/(\log x)^\beta$ for $\beta<\log4-1$.
I'm most interested in getting values of $f(x,y)$ that look like $x/(\log x)^\delta$. For example, I believe we can show (using the Selberg–Sathe formula) that $$ f\big(x, x{(\log x)^{\alpha(1-\log\alpha)-1}} \big) \sim x{(\log x)^{\alpha(1-\log\alpha)-1+\alpha\log 2+o(1)}} $$ for $\alpha\ge 2$.
Soundararajan mentions a similar result in his paper Omega results for the divisor and circle problems, but for $\sum \tau(n_j) n_j^{-3/4}$ rather than $\sum \tau(n_j)$.
Has the behavior of $f(x,y)$ already been worked out in the literature? I would very much like to cite existing results, assuming they exist, rather than prove this from scratch.
